Line data Source code
1 : /* Copyright (C) 2014 The PARI group.
2 :
3 : This file is part of the PARI/GP package.
4 :
5 : PARI/GP is free software; you can redistribute it and/or modify it under the
6 : terms of the GNU General Public License as published by the Free Software
7 : Foundation; either version 2 of the License, or (at your option) any later
8 : version. It is distributed in the hope that it will be useful, but WITHOUT
9 : ANY WARRANTY WHATSOEVER.
10 :
11 : Check the License for details. You should have received a copy of it, along
12 : with the package; see the file 'COPYING'. If not, write to the Free Software
13 : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
14 :
15 : #include "pari.h"
16 : #include "paripriv.h"
17 :
18 : #define DEBUGLEVEL DEBUGLEVEL_polmodular
19 :
20 : #define dbg_printf(lvl) if (DEBUGLEVEL >= (lvl) + 3) err_printf
21 :
22 : /**
23 : * START Code from AVSs "class_inv.h"
24 : */
25 :
26 : /* actually just returns the square-free part of the level, which is
27 : * all we care about */
28 : long
29 40281 : modinv_level(long inv)
30 : {
31 40281 : switch (inv) {
32 31731 : case INV_J: return 1;
33 1141 : case INV_G2:
34 1141 : case INV_W3W3E2:return 3;
35 1070 : case INV_F:
36 : case INV_F2:
37 : case INV_F4:
38 1070 : case INV_F8: return 6;
39 70 : case INV_F3: return 2;
40 602 : case INV_W3W3: return 6;
41 1687 : case INV_W2W7E2:
42 1687 : case INV_W2W7: return 14;
43 269 : case INV_W3W5: return 15;
44 301 : case INV_W2W3E2:
45 301 : case INV_W2W3: return 6;
46 574 : case INV_W2W5E2:
47 574 : case INV_W2W5: return 30;
48 329 : case INV_W2W13: return 26;
49 1725 : case INV_W3W7: return 42;
50 544 : case INV_W5W7: return 35;
51 56 : case INV_W3W13: return 39;
52 182 : case INV_ATKIN3: return 3;
53 : }
54 : pari_err_BUG("modinv_level"); return 0;/*LCOV_EXCL_LINE*/
55 : }
56 :
57 : /* Where applicable, returns N=p1*p2 (possibly p2=1) s.t. two j's
58 : * related to the same f are N-isogenous, and 0 otherwise. This is
59 : * often (but not necessarily) equal to the level. */
60 : long
61 7244534 : modinv_degree(long *p1, long *p2, long inv)
62 : {
63 7244534 : switch (inv) {
64 297329 : case INV_W3W5: return (*p1 = 3) * (*p2 = 5);
65 427304 : case INV_W2W3E2:
66 427304 : case INV_W2W3: return (*p1 = 2) * (*p2 = 3);
67 1537670 : case INV_W2W5E2:
68 1537670 : case INV_W2W5: return (*p1 = 2) * (*p2 = 5);
69 958287 : case INV_W2W7E2:
70 958287 : case INV_W2W7: return (*p1 = 2) * (*p2 = 7);
71 1454609 : case INV_W2W13: return (*p1 = 2) * (*p2 = 13);
72 510561 : case INV_W3W7: return (*p1 = 3) * (*p2 = 7);
73 793277 : case INV_W3W3E2:
74 793277 : case INV_W3W3: return (*p1 = 3) * (*p2 = 3);
75 349392 : case INV_W5W7: return (*p1 = 5) * (*p2 = 7);
76 195062 : case INV_W3W13: return (*p1 = 3) * (*p2 = 13);
77 163150 : case INV_ATKIN3: return (*p1 = 3) * (*p2 = 1);
78 : }
79 557893 : *p1 = *p2 = 1; return 0;
80 : }
81 :
82 : /* Certain invariants require that D not have 2 in it's conductor, but
83 : * this doesn't apply to every invariant with even level so we handle
84 : * it separately */
85 : INLINE int
86 527272 : modinv_odd_conductor(long inv)
87 : {
88 527272 : switch (inv) {
89 69164 : case INV_F:
90 : case INV_W3W3:
91 69164 : case INV_W3W7: return 1;
92 : }
93 458108 : return 0;
94 : }
95 :
96 : long
97 21310598 : modinv_height_factor(long inv)
98 : {
99 21310598 : switch (inv) {
100 5255 : case INV_J: return 1;
101 1230901 : case INV_G2: return 3;
102 3106602 : case INV_F: return 72;
103 28 : case INV_F2: return 36;
104 523481 : case INV_F3: return 24;
105 49 : case INV_F4: return 18;
106 49 : case INV_F8: return 9;
107 63 : case INV_W2W3: return 72;
108 2300998 : case INV_W3W3: return 36;
109 3606890 : case INV_W2W5: return 54;
110 1338401 : case INV_W2W7: return 48;
111 1386 : case INV_W3W5: return 36;
112 3798830 : case INV_W2W13: return 42;
113 1020635 : case INV_W3W7: return 32;
114 1132789 : case INV_W2W3E2:return 36;
115 202426 : case INV_W2W5E2:return 27;
116 976192 : case INV_W2W7E2:return 24;
117 49 : case INV_W3W3E2:return 18;
118 773948 : case INV_W5W7: return 24;
119 14 : case INV_W3W13: return 28;
120 1291612 : case INV_ATKIN3: return 2;
121 : default: pari_err_BUG("modinv_height_factor"); return 0;/*LCOV_EXCL_LINE*/
122 : }
123 : }
124 :
125 : long
126 1907423 : disc_best_modinv(long D)
127 : {
128 : long ret;
129 1907423 : ret = INV_F; if (modinv_good_disc(ret, D)) return ret;
130 1534057 : ret = INV_W2W3; if (modinv_good_disc(ret, D)) return ret;
131 1534057 : ret = INV_W2W5; if (modinv_good_disc(ret, D)) return ret;
132 1238755 : ret = INV_W2W7; if (modinv_good_disc(ret, D)) return ret;
133 1139957 : ret = INV_W2W13; if (modinv_good_disc(ret, D)) return ret;
134 838012 : ret = INV_W3W3; if (modinv_good_disc(ret, D)) return ret;
135 651805 : ret = INV_W2W3E2;if (modinv_good_disc(ret, D)) return ret;
136 579453 : ret = INV_W3W5; if (modinv_good_disc(ret, D)) return ret;
137 579299 : ret = INV_W3W7; if (modinv_good_disc(ret, D)) return ret;
138 511091 : ret = INV_W3W13; if (modinv_good_disc(ret, D)) return ret;
139 511091 : ret = INV_W2W5E2;if (modinv_good_disc(ret, D)) return ret;
140 494753 : ret = INV_F3; if (modinv_good_disc(ret, D)) return ret;
141 464485 : ret = INV_W2W7E2;if (modinv_good_disc(ret, D)) return ret;
142 376656 : ret = INV_W5W7; if (modinv_good_disc(ret, D)) return ret;
143 308581 : ret = INV_W3W3E2;if (modinv_good_disc(ret, D)) return ret;
144 308581 : ret = INV_G2; if (modinv_good_disc(ret, D)) return ret;
145 160517 : ret = INV_ATKIN3;if (modinv_good_disc(ret, D)) return ret;
146 77 : return INV_J;
147 : }
148 :
149 : INLINE long
150 44550 : modinv_sparse_factor(long inv)
151 : {
152 44550 : switch (inv) {
153 4246 : case INV_G2:
154 : case INV_F8:
155 : case INV_W3W5:
156 : case INV_W2W5E2:
157 : case INV_W3W3E2:
158 4246 : return 3;
159 583 : case INV_F:
160 583 : return 24;
161 357 : case INV_F2:
162 : case INV_W2W3:
163 357 : return 12;
164 112 : case INV_F3:
165 112 : return 8;
166 1785 : case INV_F4:
167 : case INV_W2W3E2:
168 : case INV_W2W5:
169 : case INV_W3W3:
170 1785 : return 6;
171 1046 : case INV_W2W7:
172 1046 : return 4;
173 2845 : case INV_W2W7E2:
174 : case INV_W2W13:
175 : case INV_W3W7:
176 2845 : return 2;
177 : }
178 33576 : return 1;
179 : }
180 :
181 : #define IQ_FILTER_1MOD3 1
182 : #define IQ_FILTER_2MOD3 2
183 : #define IQ_FILTER_1MOD4 4
184 : #define IQ_FILTER_3MOD4 8
185 :
186 : INLINE long
187 14938 : modinv_pfilter(long inv)
188 : {
189 14938 : switch (inv) {
190 2733 : case INV_G2:
191 : case INV_W3W3:
192 : case INV_W3W3E2:
193 : case INV_W3W5:
194 : case INV_W2W5:
195 : case INV_W2W3E2:
196 : case INV_W2W5E2:
197 : case INV_W5W7:
198 : case INV_W3W13:
199 2733 : return IQ_FILTER_1MOD3; /* ensure unique cube roots */
200 529 : case INV_W2W7:
201 : case INV_F3:
202 529 : return IQ_FILTER_1MOD4; /* ensure at most two 4th/8th roots */
203 930 : case INV_F:
204 : case INV_F2:
205 : case INV_F4:
206 : case INV_F8:
207 : case INV_W2W3:
208 : /* Ensure unique cube roots and at most two 4th/8th roots */
209 930 : return IQ_FILTER_1MOD3 | IQ_FILTER_1MOD4;
210 : }
211 10746 : return 0;
212 : }
213 :
214 : int
215 10750455 : modinv_good_prime(long inv, long p)
216 : {
217 10750455 : switch (inv) {
218 377919 : case INV_G2:
219 : case INV_W2W3E2:
220 : case INV_W3W3:
221 : case INV_W3W3E2:
222 : case INV_W3W5:
223 : case INV_W2W5E2:
224 : case INV_W2W5:
225 377919 : return (p % 3) == 2;
226 439212 : case INV_W2W7:
227 : case INV_F3:
228 439212 : return (p & 3) != 1;
229 395980 : case INV_F2:
230 : case INV_F4:
231 : case INV_F8:
232 : case INV_F:
233 : case INV_W2W3:
234 395980 : return ((p % 3) == 2) && (p & 3) != 1;
235 : }
236 9537344 : return 1;
237 : }
238 :
239 : /* Returns true if the prime p does not divide the conductor of D */
240 : INLINE int
241 3249610 : prime_to_conductor(long D, long p)
242 : {
243 : long b;
244 3249610 : if (p > 2) return (D % (p * p));
245 1244193 : b = D & 0xF;
246 1244193 : return (b && b != 4); /* 2 divides the conductor of D <=> D=0,4 mod 16 */
247 : }
248 :
249 : INLINE GEN
250 3249610 : red_primeform(long D, long p)
251 : {
252 3249610 : pari_sp av = avma;
253 : GEN P;
254 3249610 : if (!prime_to_conductor(D, p)) return NULL;
255 3249610 : P = primeform_u(stoi(D), p); /* primitive since p \nmid conductor */
256 3249610 : return gc_upto(av, qfbred_i(P));
257 : }
258 :
259 : /* Computes product of primeforms over primes appearing in the prime
260 : * factorization of n (including multiplicity) */
261 : GEN
262 135821 : qfb_nform(long D, long n)
263 : {
264 135821 : pari_sp av = avma;
265 135821 : GEN N = NULL, fa = factoru(n), P = gel(fa,1), E = gel(fa,2);
266 135821 : long i, l = lg(P);
267 :
268 407288 : for (i = 1; i < l; ++i)
269 : {
270 : long j, e;
271 271467 : GEN Q = red_primeform(D, P[i]);
272 271467 : if (!Q) return gc_NULL(av);
273 271467 : e = E[i];
274 271467 : if (i == 1) { N = Q; j = 1; } else j = 0;
275 407211 : for (; j < e; ++j) N = qfbcomp_i(Q, N);
276 : }
277 135821 : return gc_upto(av, N);
278 : }
279 :
280 : INLINE int
281 1692040 : qfb_is_two_torsion(GEN x)
282 : {
283 3384080 : return equali1(gel(x,1)) || !signe(gel(x,2))
284 3384080 : || equalii(gel(x,1), gel(x,2)) || equalii(gel(x,1), gel(x,3));
285 : }
286 :
287 : /* Returns true iff the products p1*p2, p1*p2^-1, p1^-1*p2, and
288 : * p1^-1*p2^-1 are all distinct in cl(D) */
289 : INLINE int
290 229559 : qfb_distinct_prods(long D, long p1, long p2)
291 : {
292 : GEN P1, P2;
293 :
294 229559 : P1 = red_primeform(D, p1);
295 229559 : if (!P1) return 0;
296 229559 : P1 = qfbsqr_i(P1);
297 :
298 229559 : P2 = red_primeform(D, p2);
299 229559 : if (!P2) return 0;
300 229559 : P2 = qfbsqr_i(P2);
301 :
302 229559 : return !(equalii(gel(P1,1), gel(P2,1)) && absequalii(gel(P1,2), gel(P2,2)));
303 : }
304 :
305 : /* By Corollary 3.1 of Enge-Schertz Constructing elliptic curves over finite
306 : * fields using double eta-quotients, we need p1 != p2 to both be noninert
307 : * and prime to the conductor, and if p1=p2=p we want p split and prime to the
308 : * conductor. We exclude the case that p1=p2 divides the conductor, even
309 : * though this does yield class invariants */
310 : INLINE int
311 5313964 : modinv_double_eta_good_disc(long D, long inv)
312 : {
313 5313964 : pari_sp av = avma;
314 : GEN P;
315 : long i1, i2, p1, p2, N;
316 :
317 5313964 : N = modinv_degree(&p1, &p2, inv);
318 5313964 : if (! N) return 0;
319 5313964 : i1 = kross(D, p1);
320 5313964 : if (i1 < 0) return 0;
321 : /* Exclude ramified case for w_{p,p} */
322 2406540 : if (p1 == p2 && !i1) return 0;
323 2406540 : i2 = kross(D, p2);
324 2406540 : if (i2 < 0) return 0;
325 : /* this also verifies that p1 is prime to the conductor */
326 1373203 : P = red_primeform(D, p1);
327 1373203 : if (!P || gequal1(gel(P,1)) /* don't allow p1 to be principal */
328 : /* if p1 is unramified, require it to have order > 2 */
329 1373203 : || (i1 && qfb_is_two_torsion(P))) return gc_bool(av,0);
330 1371565 : if (p1 == p2) /* if p1=p2 we need p1*p1 to be distinct from its inverse */
331 225743 : return gc_bool(av, !qfb_is_two_torsion(qfbsqr_i(P)));
332 :
333 : /* this also verifies that p2 is prime to the conductor */
334 1145822 : P = red_primeform(D, p2);
335 1145822 : if (!P || gequal1(gel(P,1)) /* don't allow p2 to be principal */
336 : /* if p2 is unramified, require it to have order > 2 */
337 1145822 : || (i2 && qfb_is_two_torsion(P))) return gc_bool(av,0);
338 1144359 : set_avma(av);
339 :
340 : /* if p1 and p2 are split, we also require p1*p2, p1*p2^-1, p1^-1*p2,
341 : * and p1^-1*p2^-1 to be distinct */
342 1144359 : if (i1>0 && i2>0 && !qfb_distinct_prods(D, p1, p2)) return gc_bool(av,0);
343 1141292 : if (!i1 && !i2) {
344 : /* if both p1 and p2 are ramified, make sure their product is not
345 : * principal */
346 135359 : P = qfb_nform(D, N);
347 135359 : if (equali1(gel(P,1))) return gc_bool(av,0);
348 135107 : set_avma(av);
349 : }
350 1141040 : return 1;
351 : }
352 :
353 : /* Assumes D is a good discriminant for inv, which implies that the
354 : * level is prime to the conductor */
355 : long
356 581 : modinv_ramified(long D, long inv, long *pN)
357 : {
358 581 : long p1, p2; *pN = modinv_degree(&p1, &p2, inv);
359 581 : if (*pN <= 1) return 0;
360 581 : return !(D % p1) && !(D % p2);
361 : }
362 :
363 : int
364 14990300 : modinv_good_disc(long inv, long D)
365 : {
366 14990300 : switch (inv) {
367 884759 : case INV_J:
368 884759 : return 1;
369 436513 : case INV_G2:
370 436513 : return !!(D % 3);
371 502845 : case INV_F3:
372 502845 : return (-D & 7) == 7;
373 2054379 : case INV_F:
374 : case INV_F2:
375 : case INV_F4:
376 : case INV_F8:
377 2054379 : return ((-D & 7) == 7) && (D % 3);
378 622069 : case INV_W3W5:
379 622069 : return (D % 3) && modinv_double_eta_good_disc(D, inv);
380 335664 : case INV_W3W3E2:
381 335664 : return (D % 3) && modinv_double_eta_good_disc(D, inv);
382 909958 : case INV_W3W3:
383 909958 : return (D & 1) && (D % 3) && modinv_double_eta_good_disc(D, inv);
384 667688 : case INV_W2W3E2:
385 667688 : return (D % 3) && modinv_double_eta_good_disc(D, inv);
386 1554721 : case INV_W2W3:
387 1554721 : return ((-D & 7) == 7) && (D % 3) && modinv_double_eta_good_disc(D, inv);
388 1581685 : case INV_W2W5:
389 1581685 : return ((-D % 80) != 20) && (D % 3) && modinv_double_eta_good_disc(D, inv);
390 540722 : case INV_W2W5E2:
391 540722 : return (D % 3) && modinv_double_eta_good_disc(D, inv);
392 576513 : case INV_W2W7E2:
393 576513 : return ((-D % 112) != 84) && modinv_double_eta_good_disc(D, inv);
394 1324607 : case INV_W2W7:
395 1324607 : return ((-D & 7) == 7) && modinv_double_eta_good_disc(D, inv);
396 1181782 : case INV_W2W13:
397 1181782 : return ((-D % 208) != 52) && modinv_double_eta_good_disc(D, inv);
398 666806 : case INV_W3W7:
399 666806 : return (D & 1) && (-D % 21) && modinv_double_eta_good_disc(D, inv);
400 450975 : case INV_W5W7: /* NB: This is a guess; avs doesn't have an entry */
401 450975 : return (D % 3) && modinv_double_eta_good_disc(D, inv);
402 520688 : case INV_W3W13: /* NB: This is a guess; avs doesn't have an entry */
403 520688 : return (D & 1) && (D % 3) && modinv_double_eta_good_disc(D, inv);
404 177926 : case INV_ATKIN3:
405 177926 : return (D%3!=2 && D%9 && (D<-36 || D==-15 || D==-23 || D==-24));
406 : }
407 0 : pari_err_BUG("modinv_good_discriminant");
408 : return 0;/*LCOV_EXCL_LINE*/
409 : }
410 :
411 : int
412 945 : modinv_is_Weber(long inv)
413 : {
414 0 : return inv == INV_F || inv == INV_F2 || inv == INV_F3 || inv == INV_F4
415 945 : || inv == INV_F8;
416 : }
417 :
418 : int
419 236998 : modinv_is_double_eta(long inv)
420 : {
421 236998 : switch (inv) {
422 34637 : case INV_W2W3:
423 : case INV_W2W3E2:
424 : case INV_W2W5:
425 : case INV_W2W5E2:
426 : case INV_W2W7:
427 : case INV_W2W7E2:
428 : case INV_W2W13:
429 : case INV_W3W3:
430 : case INV_W3W3E2:
431 : case INV_W3W5:
432 : case INV_W3W7:
433 : case INV_W5W7:
434 : case INV_W3W13:
435 : case INV_ATKIN3: /* as far as we are concerned */
436 34637 : return 1;
437 : }
438 202361 : return 0;
439 : }
440 :
441 : /* END Code from "class_inv.h" */
442 :
443 : INLINE int
444 10201 : safe_abs_sqrt(ulong *r, ulong x, ulong p, ulong pi, ulong s2)
445 : {
446 10201 : if (krouu(x, p) == -1)
447 : {
448 4622 : if (p%4 == 1) return 0;
449 4622 : x = Fl_neg(x, p);
450 : }
451 10201 : *r = Fl_sqrt_pre_i(x, s2, p, pi);
452 10201 : return 1;
453 : }
454 :
455 : INLINE int
456 5053 : eighth_root(ulong *r, ulong x, ulong p, ulong pi, ulong s2)
457 : {
458 : ulong s;
459 5053 : if (krouu(x, p) == -1) return 0;
460 2812 : s = Fl_sqrt_pre_i(x, s2, p, pi);
461 2812 : return safe_abs_sqrt(&s, s, p, pi, s2) && safe_abs_sqrt(r, s, p, pi, s2);
462 : }
463 :
464 : INLINE ulong
465 3070 : modinv_f_from_j(ulong j, ulong p, ulong pi, ulong s2, long only_residue)
466 : {
467 3070 : pari_sp av = avma;
468 : GEN pol, r;
469 : long i;
470 3070 : ulong g2, f = ULONG_MAX;
471 :
472 : /* f^8 must be a root of X^3 - \gamma_2 X - 16 */
473 3070 : g2 = Fl_sqrtl_pre(j, 3, p, pi);
474 :
475 3070 : pol = mkvecsmall5(0UL, Fl_neg(16 % p, p), Fl_neg(g2, p), 0UL, 1UL);
476 3070 : r = Flx_roots_pre(pol, p, pi);
477 5586 : for (i = 1; i < lg(r); ++i)
478 5586 : if (only_residue)
479 1248 : { if (krouu(r[i], p) != -1) return gc_ulong(av,r[i]); }
480 4338 : else if (eighth_root(&f, r[i], p, pi, s2)) return gc_ulong(av,f);
481 0 : pari_err_BUG("modinv_f_from_j");
482 : return 0;/*LCOV_EXCL_LINE*/
483 : }
484 :
485 : INLINE ulong
486 358 : modinv_f3_from_j(ulong j, ulong p, ulong pi, ulong s2)
487 : {
488 358 : pari_sp av = avma;
489 : GEN pol, r;
490 : long i;
491 358 : ulong f = ULONG_MAX;
492 :
493 358 : pol = mkvecsmall5(0UL,
494 358 : Fl_neg(4096 % p, p), Fl_sub(768 % p, j, p), Fl_neg(48 % p, p), 1UL);
495 358 : r = Flx_roots_pre(pol, p, pi);
496 715 : for (i = 1; i < lg(r); ++i)
497 715 : if (eighth_root(&f, r[i], p, pi, s2)) return gc_ulong(av,f);
498 0 : pari_err_BUG("modinv_f3_from_j");
499 : return 0;/*LCOV_EXCL_LINE*/
500 : }
501 :
502 : /* Return the exponent e for the double-eta "invariant" w such that
503 : * w^e is a class invariant. For example w2w3^12 is a class
504 : * invariant, so double_eta_exponent(INV_W2W3) is 12 and
505 : * double_eta_exponent(INV_W2W3E2) is 6. */
506 : INLINE ulong
507 58637 : double_eta_exponent(long inv)
508 : {
509 58637 : switch (inv) {
510 2452 : case INV_W2W3: return 12;
511 14666 : case INV_W2W3E2:
512 : case INV_W2W5:
513 14666 : case INV_W3W3: return 6;
514 10147 : case INV_W2W7: return 4;
515 5406 : case INV_W3W5:
516 : case INV_W2W5E2:
517 5406 : case INV_W3W3E2: return 3;
518 14684 : case INV_W2W7E2:
519 : case INV_W2W13:
520 14684 : case INV_W3W7: return 2;
521 11282 : default: return 1;
522 : }
523 : }
524 :
525 : INLINE ulong
526 63 : weber_exponent(long inv)
527 : {
528 63 : switch (inv)
529 : {
530 56 : case INV_F: return 24;
531 0 : case INV_F2: return 12;
532 7 : case INV_F3: return 8;
533 0 : case INV_F4: return 6;
534 0 : case INV_F8: return 3;
535 0 : default: return 1;
536 : }
537 : }
538 :
539 : INLINE ulong
540 30911 : double_eta_power(long inv, ulong w, ulong p, ulong pi)
541 : {
542 30911 : return Fl_powu_pre(w, double_eta_exponent(inv), p, pi);
543 : }
544 :
545 : static GEN
546 231 : double_eta_raw_to_Fp(GEN f, GEN p)
547 : {
548 231 : GEN u = FpX_red(RgV_to_RgX(gel(f,1), 0), p);
549 231 : GEN v = FpX_red(RgV_to_RgX(gel(f,2), 0), p);
550 231 : return mkvec3(u, v, gel(f,3));
551 : }
552 :
553 : /* Given a root x of polclass(D, inv) modulo N, returns a root of polclass(D,0)
554 : * modulo N by plugging x to a modular polynomial. For double-eta quotients,
555 : * this is done by plugging x into the modular polynomial Phi(INV_WpWq, j)
556 : * Enge, Morain 2013: Generalised Weber Functions. */
557 : GEN
558 1021 : Fp_modinv_to_j(GEN x, long inv, GEN p)
559 : {
560 1021 : switch(inv)
561 : {
562 363 : case INV_J: return Fp_red(x, p);
563 364 : case INV_G2: return Fp_powu(x, 3, p);
564 63 : case INV_F: case INV_F2: case INV_F3: case INV_F4: case INV_F8:
565 : {
566 63 : GEN xe = Fp_powu(x, weber_exponent(inv), p);
567 63 : return Fp_div(Fp_powu(subiu(xe, 16), 3, p), xe, p);
568 : }
569 231 : default:
570 231 : if (modinv_is_double_eta(inv))
571 : {
572 231 : GEN xe = Fp_powu(x, double_eta_exponent(inv), p);
573 231 : GEN uvk = double_eta_raw_to_Fp(double_eta_raw(inv), p);
574 231 : GEN J0 = FpX_eval(gel(uvk,1), xe, p);
575 231 : GEN J1 = FpX_eval(gel(uvk,2), xe, p);
576 231 : GEN J2 = Fp_pow(xe, gel(uvk,3), p);
577 231 : GEN phi = mkvec3(J0, J1, J2);
578 231 : return FpX_oneroot(RgX_to_FpX(RgV_to_RgX(phi,1), p),p);
579 : }
580 : pari_err_BUG("Fp_modinv_to_j"); return NULL;/* LCOV_EXCL_LINE */
581 : }
582 : }
583 :
584 : /* Assuming p = 2 (mod 3) and p = 3 (mod 4): if the two 12th roots of
585 : * x (mod p) exist, set *r to one of them and return 1, otherwise
586 : * return 0 (without touching *r). */
587 : INLINE int
588 899 : twelth_root(ulong *r, ulong x, ulong p, ulong pi, ulong s2)
589 : {
590 899 : ulong t = Fl_sqrtl_pre(x, 3, p, pi);
591 899 : if (krouu(t, p) == -1) return 0;
592 850 : t = Fl_sqrt_pre_i(t, s2, p, pi);
593 850 : return safe_abs_sqrt(r, t, p, pi, s2);
594 : }
595 :
596 : INLINE int
597 6153 : sixth_root(ulong *r, ulong x, ulong p, ulong pi, ulong s2)
598 : {
599 6153 : ulong t = Fl_sqrtl_pre(x, 3, p, pi);
600 6153 : if (krouu(t, p) == -1) return 0;
601 5941 : *r = Fl_sqrt_pre_i(t, s2, p, pi);
602 5941 : return 1;
603 : }
604 :
605 : INLINE int
606 4073 : fourth_root(ulong *r, ulong x, ulong p, ulong pi, ulong s2)
607 : {
608 : ulong s;
609 4073 : if (krouu(x, p) == -1) return 0;
610 3727 : s = Fl_sqrt_pre_i(x, s2, p, pi);
611 3727 : return safe_abs_sqrt(r, s, p, pi, s2);
612 : }
613 :
614 : INLINE int
615 27495 : double_eta_root(long inv, ulong *r, ulong w, ulong p, ulong pi, ulong s2)
616 : {
617 27495 : switch (double_eta_exponent(inv)) {
618 899 : case 12: return twelth_root(r, w, p, pi, s2);
619 6153 : case 6: return sixth_root(r, w, p, pi, s2);
620 4073 : case 4: return fourth_root(r, w, p, pi, s2);
621 2330 : case 3: *r = Fl_sqrtl_pre(w, 3, p, pi); return 1;
622 7846 : case 2: return krouu(w, p) != -1 && !!(*r = Fl_sqrt_pre_i(w, s2, p, pi));
623 6194 : default: *r = w; return 1; /* case 1 */
624 : }
625 : }
626 :
627 : /* F = double_eta_Fl(inv, p) */
628 : static GEN
629 47105 : Flx_double_eta_xpoly(GEN F, ulong j, ulong p, ulong pi)
630 : {
631 47105 : GEN u = gel(F,1), v = gel(F,2), w;
632 47105 : long i, k = itos(gel(F,3)), lu = lg(u), lv = lg(v), lw = lu + 1;
633 :
634 47105 : w = cgetg(lw, t_VECSMALL); /* lu >= max(lv,k) */
635 47106 : w[1] = 0; /* variable number */
636 1154737 : for (i = 1; i < lv; i++) uel(w, i+1) = Fl_add(uel(u,i), Fl_mul_pre(j, uel(v,i), p, pi), p);
637 94218 : for ( ; i < lu; i++) uel(w, i+1) = uel(u,i);
638 47109 : uel(w, k+2) = Fl_add(uel(w, k+2), Fl_sqr_pre(j, p, pi), p);
639 47108 : return Flx_renormalize(w, lw);
640 : }
641 :
642 : /* F = double_eta_Fl(inv, p) */
643 : static GEN
644 30911 : Flx_double_eta_jpoly(GEN F, ulong x, ulong p, ulong pi)
645 : {
646 30911 : pari_sp av = avma;
647 30911 : GEN u = gel(F,1), v = gel(F,2), xs;
648 30911 : long k = itos(gel(F,3));
649 : ulong a, b, c;
650 :
651 : /* u is always longest and the length is bigger than k */
652 30911 : xs = Fl_powers_pre(x, lg(u) - 1, p, pi);
653 30911 : c = Flv_dotproduct_pre(u, xs, p, pi);
654 30911 : b = Flv_dotproduct_pre(v, xs, p, pi);
655 30911 : a = uel(xs, k + 1);
656 30911 : set_avma(av);
657 30911 : return mkvecsmall4(0, c, b, a);
658 : }
659 :
660 : /* reduce F = double_eta_raw(inv) mod p */
661 : static GEN
662 33047 : double_eta_raw_to_Fl(GEN f, ulong p)
663 : {
664 33047 : GEN u = ZV_to_Flv(gel(f,1), p);
665 33047 : GEN v = ZV_to_Flv(gel(f,2), p);
666 33046 : return mkvec3(u, v, gel(f,3));
667 : }
668 : /* double_eta_raw(inv) mod p */
669 : static GEN
670 26535 : double_eta_Fl(long inv, ulong p)
671 26535 : { return double_eta_raw_to_Fl(double_eta_raw(inv), p); }
672 :
673 : /* Go through roots of Psi(X,j) until one has an double_eta_exponent(inv)-th
674 : * root, and return that root. F = double_eta_Fl(inv,p) */
675 : INLINE ulong
676 5963 : modinv_double_eta_from_j(GEN F, long inv, ulong j, ulong p, ulong pi, ulong s2)
677 : {
678 5963 : pari_sp av = avma;
679 : long i;
680 5963 : ulong f = ULONG_MAX;
681 5963 : GEN a = Flx_double_eta_xpoly(F, j, p, pi);
682 5962 : a = Flx_roots_pre(a, p, pi);
683 6922 : for (i = 1; i < lg(a); ++i)
684 6922 : if (double_eta_root(inv, &f, uel(a, i), p, pi, s2)) break;
685 5963 : if (i == lg(a)) pari_err_BUG("modinv_double_eta_from_j");
686 5963 : return gc_ulong(av,f);
687 : }
688 :
689 : /* assume j1 != j2 */
690 : static long
691 14609 : modinv_double_eta_from_2j(
692 : ulong *r, long inv, ulong j1, ulong j2, ulong p, ulong pi, ulong s2)
693 : {
694 14609 : GEN f, g, d, F = double_eta_Fl(inv, p);
695 14609 : f = Flx_double_eta_xpoly(F, j1, p, pi);
696 14609 : g = Flx_double_eta_xpoly(F, j2, p, pi);
697 14609 : d = Flx_gcd(f, g, p);
698 : /* we should have deg(d) = 1, but because j1 or j2 may not have the correct
699 : * endomorphism ring, we use the less strict conditional underneath */
700 29220 : return (degpol(d) > 2 || (*r = Flx_oneroot_pre(d, p, pi)) == p
701 29220 : || ! double_eta_root(inv, r, *r, p, pi, s2));
702 : }
703 :
704 : long
705 14688 : modfn_unambiguous_root(ulong *r, long inv, ulong j0, norm_eqn_t ne, GEN jdb)
706 : {
707 14688 : pari_sp av = avma;
708 14688 : long p1, p2, v = ne->v, p1_depth;
709 14688 : ulong j1, p = ne->p, pi = ne->pi, s2 = ne->s2;
710 : GEN phi;
711 :
712 14688 : (void) modinv_degree(&p1, &p2, inv);
713 14688 : p1_depth = u_lval(v, p1);
714 :
715 14688 : phi = polmodular_db_getp(jdb, p1, p);
716 14688 : if (!next_surface_nbr(&j1, phi, p1, p1_depth, j0, NULL, p, pi))
717 0 : pari_err_BUG("modfn_unambiguous_root");
718 14688 : if (p2 == p1) {
719 2354 : if (!next_surface_nbr(&j1, phi, p1, p1_depth, j1, &j0, p, pi))
720 0 : pari_err_BUG("modfn_unambiguous_root");
721 12334 : } else if (p2 > 1)
722 : {
723 9806 : long p2_depth = u_lval(v, p2);
724 9806 : phi = polmodular_db_getp(jdb, p2, p);
725 9805 : if (!next_surface_nbr(&j1, phi, p2, p2_depth, j1, NULL, p, pi))
726 0 : pari_err_BUG("modfn_unambiguous_root");
727 : }
728 16802 : return gc_long(av, j1 != j0
729 14679 : && !modinv_double_eta_from_2j(r, inv, j0, j1, p, pi, s2));
730 : }
731 :
732 : ulong
733 192129 : modfn_root(ulong j, norm_eqn_t ne, long inv)
734 : {
735 192129 : ulong f, p = ne->p, pi = ne->pi, s2 = ne->s2;
736 192129 : switch (inv) {
737 182115 : case INV_J: return j;
738 6586 : case INV_G2: return Fl_sqrtl_pre(j, 3, p, pi);
739 1705 : case INV_F: return modinv_f_from_j(j, p, pi, s2, 0);
740 196 : case INV_F2:
741 196 : f = modinv_f_from_j(j, p, pi, s2, 0);
742 196 : return Fl_sqr_pre(f, p, pi);
743 358 : case INV_F3: return modinv_f3_from_j(j, p, pi, s2);
744 553 : case INV_F4:
745 553 : f = modinv_f_from_j(j, p, pi, s2, 0);
746 553 : return Fl_sqr_pre(Fl_sqr_pre(f, p, pi), p, pi);
747 616 : case INV_F8: return modinv_f_from_j(j, p, pi, s2, 1);
748 : }
749 0 : if (modinv_is_double_eta(inv))
750 : {
751 0 : pari_sp av = avma;
752 0 : ulong f = modinv_double_eta_from_j(double_eta_Fl(inv,p), inv, j, p, pi, s2);
753 0 : return gc_ulong(av,f);
754 : }
755 : pari_err_BUG("modfn_root"); return ULONG_MAX;/*LCOV_EXCL_LINE*/
756 : }
757 :
758 : /* F = double_eta_raw(inv) */
759 : long
760 6511 : modinv_j_from_2double_eta(
761 : GEN F, long inv, ulong x0, ulong x1, ulong p, ulong pi)
762 : {
763 : GEN f, g, d;
764 :
765 6511 : x0 = double_eta_power(inv, x0, p, pi);
766 6511 : x1 = double_eta_power(inv, x1, p, pi);
767 6511 : F = double_eta_raw_to_Fl(F, p);
768 6511 : f = Flx_double_eta_jpoly(F, x0, p, pi);
769 6511 : g = Flx_double_eta_jpoly(F, x1, p, pi);
770 6511 : d = Flx_gcd(f, g, p); /* >= 1 */
771 6511 : return degpol(d) == 1;
772 : }
773 :
774 : /* x root of (X^24 - 16)^3 - X^24 * j = 0 => j = (x^24 - 16)^3 / x^24 */
775 : INLINE ulong
776 1830 : modinv_j_from_f(ulong x, ulong n, ulong p, ulong pi)
777 : {
778 1830 : ulong x24 = Fl_powu_pre(x, 24 / n, p, pi);
779 1830 : return Fl_div(Fl_powu_pre(Fl_sub(x24, 16 % p, p), 3, p, pi), x24, p);
780 : }
781 : /* should never be called if modinv_double_eta(inv) is true */
782 : INLINE ulong
783 65451 : modfn_preimage(ulong x, ulong p, ulong pi, long inv)
784 : {
785 65451 : switch (inv) {
786 58687 : case INV_J: return x;
787 4934 : case INV_G2: return Fl_powu_pre(x, 3, p, pi);
788 : /* NB: could replace these with a single call modinv_j_from_f(x,inv,p,pi)
789 : * but avoid the dependence on the actual value of inv */
790 626 : case INV_F: return modinv_j_from_f(x, 1, p, pi);
791 196 : case INV_F2: return modinv_j_from_f(x, 2, p, pi);
792 168 : case INV_F3: return modinv_j_from_f(x, 3, p, pi);
793 392 : case INV_F4: return modinv_j_from_f(x, 4, p, pi);
794 448 : case INV_F8: return modinv_j_from_f(x, 8, p, pi);
795 : }
796 : pari_err_BUG("modfn_preimage"); return ULONG_MAX;/*LCOV_EXCL_LINE*/
797 : }
798 :
799 : /* SECTION: class group bb_group. */
800 :
801 : INLINE GEN
802 134944 : mkqfis(GEN a, ulong b, ulong c, GEN D) { retmkqfb(a, utoi(b), utoi(c), D); }
803 :
804 : /* SECTION: dot-product-like functions on Fl's with precomputed inverse. */
805 :
806 : /* Computes x0y1 + y0x1 (mod p); assumes p < 2^63. */
807 : INLINE ulong
808 55874937 : Fl_addmul2(
809 : ulong x0, ulong x1, ulong y0, ulong y1,
810 : ulong p, ulong pi)
811 : {
812 55874937 : return Fl_addmulmul_pre(x0,y1,y0,x1,p,pi);
813 : }
814 :
815 : /* Computes x0y2 + x1y1 + x2y0 (mod p); assumes p < 2^62. */
816 : INLINE ulong
817 9721088 : Fl_addmul3(
818 : ulong x0, ulong x1, ulong x2, ulong y0, ulong y1, ulong y2,
819 : ulong p, ulong pi)
820 : {
821 : ulong l0, l1, h0, h1;
822 : LOCAL_OVERFLOW;
823 : LOCAL_HIREMAINDER;
824 9721088 : l0 = mulll(x0, y2); h0 = hiremainder;
825 9721088 : l1 = mulll(x1, y1); h1 = hiremainder;
826 9721088 : l1 = addll(l0, l1); h1 = addllx(h0, h1);
827 9721088 : l0 = mulll(x2, y0); h0 = hiremainder;
828 9721088 : l1 = addll(l0, l1); h1 = addllx(h0, h1);
829 9721088 : return remll_pre(h1, l1, p, pi);
830 : }
831 :
832 : /* Computes x0y3 + x1y2 + x2y1 + x3y0 (mod p); assumes p < 2^62. */
833 : INLINE ulong
834 5021644 : Fl_addmul4(
835 : ulong x0, ulong x1, ulong x2, ulong x3,
836 : ulong y0, ulong y1, ulong y2, ulong y3,
837 : ulong p, ulong pi)
838 : {
839 : ulong l0, l1, h0, h1;
840 : LOCAL_OVERFLOW;
841 : LOCAL_HIREMAINDER;
842 5021644 : l0 = mulll(x0, y3); h0 = hiremainder;
843 5021644 : l1 = mulll(x1, y2); h1 = hiremainder;
844 5021644 : l1 = addll(l0, l1); h1 = addllx(h0, h1);
845 5021644 : l0 = mulll(x2, y1); h0 = hiremainder;
846 5021644 : l1 = addll(l0, l1); h1 = addllx(h0, h1);
847 5021644 : l0 = mulll(x3, y0); h0 = hiremainder;
848 5021644 : l1 = addll(l0, l1); h1 = addllx(h0, h1);
849 5021644 : return remll_pre(h1, l1, p, pi);
850 : }
851 :
852 : /* Computes x0y4 + x1y3 + x2y2 + x3y1 + x4y0 (mod p); assumes p < 2^62. */
853 : INLINE ulong
854 24980647 : Fl_addmul5(
855 : ulong x0, ulong x1, ulong x2, ulong x3, ulong x4,
856 : ulong y0, ulong y1, ulong y2, ulong y3, ulong y4,
857 : ulong p, ulong pi)
858 : {
859 : ulong l0, l1, h0, h1;
860 : LOCAL_OVERFLOW;
861 : LOCAL_HIREMAINDER;
862 24980647 : l0 = mulll(x0, y4); h0 = hiremainder;
863 24980647 : l1 = mulll(x1, y3); h1 = hiremainder;
864 24980647 : l1 = addll(l0, l1); h1 = addllx(h0, h1);
865 24980647 : l0 = mulll(x2, y2); h0 = hiremainder;
866 24980647 : l1 = addll(l0, l1); h1 = addllx(h0, h1);
867 24980647 : l0 = mulll(x3, y1); h0 = hiremainder;
868 24980647 : l1 = addll(l0, l1); h1 = addllx(h0, h1);
869 24980647 : l0 = mulll(x4, y0); h0 = hiremainder;
870 24980647 : l1 = addll(l0, l1); h1 = addllx(h0, h1);
871 24980647 : return remll_pre(h1, l1, p, pi);
872 : }
873 :
874 : /* A polmodular database for a given class invariant consists of a t_VEC whose
875 : * L-th entry is 0 or a GEN pointing to Phi_L. This function produces a pair
876 : * of databases corresponding to the j-invariant and inv */
877 : GEN
878 21471 : polmodular_db_init(long inv)
879 : {
880 21471 : const long LEN = 32;
881 21471 : GEN res = cgetg_block(3, t_VEC);
882 21471 : gel(res, 1) = zerovec_block(LEN);
883 21471 : gel(res, 2) = (inv == INV_J)? gen_0: zerovec_block(LEN);
884 21471 : return res;
885 : }
886 :
887 : void
888 25057 : polmodular_db_add_level(GEN *DB, long L, long inv)
889 : {
890 25057 : GEN db = gel(*DB, (inv == INV_J)? 1: 2);
891 25057 : long max_L = lg(db) - 1;
892 25057 : if (L > max_L) {
893 : GEN newdb;
894 43 : long i, newlen = 2 * L;
895 :
896 43 : newdb = cgetg_block(newlen + 1, t_VEC);
897 1419 : for (i = 1; i <= max_L; ++i) gel(newdb, i) = gel(db, i);
898 2941 : for ( ; i <= newlen; ++i) gel(newdb, i) = gen_0;
899 43 : killblock(db);
900 43 : gel(*DB, (inv == INV_J)? 1: 2) = db = newdb;
901 : }
902 25057 : if (typ(gel(db, L)) == t_INT) {
903 8275 : pari_sp av = avma;
904 8275 : GEN x = polmodular0_ZM(L, inv, NULL, NULL, 0, DB); /* may set db[L] */
905 8275 : GEN y = gel(db, L);
906 8275 : gel(db, L) = gclone(x);
907 8275 : if (typ(y) != t_INT) gunclone(y);
908 8275 : set_avma(av);
909 : }
910 25057 : }
911 :
912 : void
913 4913 : polmodular_db_add_levels(GEN *db, long *levels, long k, long inv)
914 : {
915 : long i;
916 10263 : for (i = 0; i < k; ++i) polmodular_db_add_level(db, levels[i], inv);
917 4913 : }
918 :
919 : GEN
920 356906 : polmodular_db_for_inv(GEN db, long inv) { return gel(db, (inv==INV_J)? 1: 2); }
921 :
922 : /* TODO: Also cache modpoly mod p for most recent p (avoid repeated
923 : * reductions in, for example, polclass.c:oneroot_of_classpoly(). */
924 : GEN
925 519528 : polmodular_db_getp(GEN db, long L, ulong p)
926 : {
927 519528 : GEN f = gel(db, L);
928 519528 : if (isintzero(f)) pari_err_BUG("polmodular_db_getp");
929 519528 : return ZM_to_Flm(f, p);
930 : }
931 :
932 : /* SECTION: Table of discriminants to use. */
933 : typedef struct {
934 : long GENcode0; /* used when serializing the struct to a t_VECSMALL */
935 : long inv; /* invariant */
936 : long L; /* modpoly level */
937 : long D0; /* fundamental discriminant */
938 : long D1; /* chosen discriminant */
939 : long L0; /* first generator norm */
940 : long L1; /* second generator norm */
941 : long n1; /* order of L0 in cl(D1) */
942 : long n2; /* order of L0 in cl(D2) where D2 = L^2 D1 */
943 : long dl1; /* m such that L0^m = L in cl(D1) */
944 : long dl2_0; /* These two are (m, n) such that L0^m L1^n = form of norm L^2 in D2 */
945 : long dl2_1; /* This n is always 1 or 0. */
946 : /* this part is not serialized */
947 : long nprimes; /* number of primes needed for D1 */
948 : long cost; /* cost to enumerate subgroup of cl(L^2D): subgroup size is n2 if L1=0, 2*n2 o.w. */
949 : long bits;
950 : ulong *primes;
951 : ulong *traces;
952 : } disc_info;
953 :
954 : #define MODPOLY_MAX_DCNT 64
955 :
956 : /* Flag for last parameter of discriminant_with_classno_at_least.
957 : * Warning: ignoring the sparse factor makes everything slower by
958 : * something like (sparse factor)^3. */
959 : #define USE_SPARSE_FACTOR 0
960 : #define IGNORE_SPARSE_FACTOR 1
961 :
962 : static long
963 : discriminant_with_classno_at_least(disc_info Ds[MODPOLY_MAX_DCNT], long L,
964 : long inv, GEN Q, long ignore_sparse);
965 :
966 : /* SECTION: evaluation functions for modular polynomials of small level. */
967 :
968 : /* Based on phi2_eval_ff() in Sutherland's classpoly programme.
969 : * Calculates Phi_2(X, j) (mod p) with 6M+7A (4 reductions, not
970 : * counting those for Phi_2) */
971 : INLINE GEN
972 26399431 : Flm_Fl_phi2_evalx(GEN phi2, ulong j, ulong p, ulong pi)
973 : {
974 26399431 : GEN res = cgetg(6, t_VECSMALL);
975 : ulong j2, t1;
976 :
977 26362843 : res[1] = 0; /* variable name */
978 :
979 26362843 : j2 = Fl_sqr_pre(j, p, pi);
980 26376903 : t1 = Fl_add(j, coeff(phi2, 3, 1), p);
981 26377997 : t1 = Fl_addmul2(j, j2, t1, coeff(phi2, 2, 1), p, pi);
982 26434640 : res[2] = Fl_add(t1, coeff(phi2, 1, 1), p);
983 :
984 26407226 : t1 = Fl_addmul2(j, j2, coeff(phi2, 3, 2), coeff(phi2, 2, 2), p, pi);
985 26446007 : res[3] = Fl_add(t1, coeff(phi2, 2, 1), p);
986 :
987 26420004 : t1 = Fl_mul_pre(j, coeff(phi2, 3, 2), p, pi);
988 26434171 : t1 = Fl_add(t1, coeff(phi2, 3, 1), p);
989 26414670 : res[4] = Fl_sub(t1, j2, p);
990 :
991 26389357 : res[5] = 1;
992 26389357 : return res;
993 : }
994 :
995 : /* Based on phi3_eval_ff() in Sutherland's classpoly programme.
996 : * Calculates Phi_3(X, j) (mod p) with 13M+13A (6 reductions, not
997 : * counting those for Phi_3) */
998 : INLINE GEN
999 3242686 : Flm_Fl_phi3_evalx(GEN phi3, ulong j, ulong p, ulong pi)
1000 : {
1001 3242686 : GEN res = cgetg(7, t_VECSMALL);
1002 : ulong j2, j3, t1;
1003 :
1004 3241168 : res[1] = 0; /* variable name */
1005 :
1006 3241168 : j2 = Fl_sqr_pre(j, p, pi);
1007 3243309 : j3 = Fl_mul_pre(j, j2, p, pi);
1008 :
1009 3244114 : t1 = Fl_add(j, coeff(phi3, 4, 1), p);
1010 6491370 : res[2] = Fl_addmul3(j, j2, j3, t1,
1011 3244238 : coeff(phi3, 3, 1), coeff(phi3, 2, 1), p, pi);
1012 :
1013 3247132 : t1 = Fl_addmul3(j, j2, j3, coeff(phi3, 4, 2),
1014 3247132 : coeff(phi3, 3, 2), coeff(phi3, 2, 2), p, pi);
1015 3246807 : res[3] = Fl_add(t1, coeff(phi3, 2, 1), p);
1016 :
1017 3245558 : t1 = Fl_addmul3(j, j2, j3, coeff(phi3, 4, 3),
1018 3245558 : coeff(phi3, 3, 3), coeff(phi3, 3, 2), p, pi);
1019 3247615 : res[4] = Fl_add(t1, coeff(phi3, 3, 1), p);
1020 :
1021 3246447 : t1 = Fl_addmul2(j, j2, coeff(phi3, 4, 3), coeff(phi3, 4, 2), p, pi);
1022 3246996 : t1 = Fl_add(t1, coeff(phi3, 4, 1), p);
1023 3245945 : res[5] = Fl_sub(t1, j3, p);
1024 :
1025 3245071 : res[6] = 1;
1026 3245071 : return res;
1027 : }
1028 :
1029 : /* Based on phi5_eval_ff() in Sutherland's classpoly programme.
1030 : * Calculates Phi_5(X, j) (mod p) with 33M+31A (10 reductions, not
1031 : * counting those for Phi_5) */
1032 : INLINE GEN
1033 5012513 : Flm_Fl_phi5_evalx(GEN phi5, ulong j, ulong p, ulong pi)
1034 : {
1035 5012513 : GEN res = cgetg(9, t_VECSMALL);
1036 : ulong j2, j3, j4, j5, t1;
1037 :
1038 5008892 : res[1] = 0; /* variable name */
1039 :
1040 5008892 : j2 = Fl_sqr_pre(j, p, pi);
1041 5012507 : j3 = Fl_mul_pre(j, j2, p, pi);
1042 5014128 : j4 = Fl_sqr_pre(j2, p, pi);
1043 5013777 : j5 = Fl_mul_pre(j, j4, p, pi);
1044 :
1045 5016346 : t1 = Fl_add(j, coeff(phi5, 6, 1), p);
1046 5014816 : t1 = Fl_addmul5(j, j2, j3, j4, j5, t1,
1047 5014816 : coeff(phi5, 5, 1), coeff(phi5, 4, 1),
1048 5014816 : coeff(phi5, 3, 1), coeff(phi5, 2, 1),
1049 : p, pi);
1050 5022933 : res[2] = Fl_add(t1, coeff(phi5, 1, 1), p);
1051 :
1052 5020227 : t1 = Fl_addmul5(j, j2, j3, j4, j5,
1053 5020227 : coeff(phi5, 6, 2), coeff(phi5, 5, 2),
1054 5020227 : coeff(phi5, 4, 2), coeff(phi5, 3, 2), coeff(phi5, 2, 2),
1055 : p, pi);
1056 5023252 : res[3] = Fl_add(t1, coeff(phi5, 2, 1), p);
1057 :
1058 5019705 : t1 = Fl_addmul5(j, j2, j3, j4, j5,
1059 5019705 : coeff(phi5, 6, 3), coeff(phi5, 5, 3),
1060 5019705 : coeff(phi5, 4, 3), coeff(phi5, 3, 3), coeff(phi5, 3, 2),
1061 : p, pi);
1062 5024515 : res[4] = Fl_add(t1, coeff(phi5, 3, 1), p);
1063 :
1064 5020821 : t1 = Fl_addmul5(j, j2, j3, j4, j5,
1065 5020821 : coeff(phi5, 6, 4), coeff(phi5, 5, 4),
1066 5020821 : coeff(phi5, 4, 4), coeff(phi5, 4, 3), coeff(phi5, 4, 2),
1067 : p, pi);
1068 5024815 : res[5] = Fl_add(t1, coeff(phi5, 4, 1), p);
1069 :
1070 5021097 : t1 = Fl_addmul5(j, j2, j3, j4, j5,
1071 5021097 : coeff(phi5, 6, 5), coeff(phi5, 5, 5),
1072 5021097 : coeff(phi5, 5, 4), coeff(phi5, 5, 3), coeff(phi5, 5, 2),
1073 : p, pi);
1074 5026000 : res[6] = Fl_add(t1, coeff(phi5, 5, 1), p);
1075 :
1076 5022782 : t1 = Fl_addmul4(j, j2, j3, j4,
1077 5022782 : coeff(phi5, 6, 5), coeff(phi5, 6, 4),
1078 5022782 : coeff(phi5, 6, 3), coeff(phi5, 6, 2),
1079 : p, pi);
1080 5026539 : t1 = Fl_add(t1, coeff(phi5, 6, 1), p);
1081 5022892 : res[7] = Fl_sub(t1, j5, p);
1082 :
1083 5020272 : res[8] = 1;
1084 5020272 : return res;
1085 : }
1086 :
1087 : GEN
1088 41851769 : Flm_Fl_polmodular_evalx(GEN phi, long L, ulong j, ulong p, ulong pi)
1089 : {
1090 41851769 : switch (L) {
1091 26402104 : case 2: return Flm_Fl_phi2_evalx(phi, j, p, pi);
1092 3242136 : case 3: return Flm_Fl_phi3_evalx(phi, j, p, pi);
1093 5011941 : case 5: return Flm_Fl_phi5_evalx(phi, j, p, pi);
1094 7195588 : default: { /* not GC clean, but gc_upto-safe */
1095 7195588 : GEN j_powers = Fl_powers_pre(j, L + 1, p, pi);
1096 7262209 : return Flm_Flc_mul_pre_Flx(phi, j_powers, p, pi, 0);
1097 : }
1098 : }
1099 : }
1100 :
1101 : /* SECTION: Velu's formula for the codmain curve (Fl case). */
1102 :
1103 : INLINE ulong
1104 1684050 : Fl_mul4(ulong x, ulong p)
1105 1684050 : { return Fl_double(Fl_double(x, p), p); }
1106 :
1107 : INLINE ulong
1108 91883 : Fl_mul5(ulong x, ulong p)
1109 91883 : { return Fl_add(x, Fl_mul4(x, p), p); }
1110 :
1111 : INLINE ulong
1112 842063 : Fl_mul8(ulong x, ulong p)
1113 842063 : { return Fl_double(Fl_mul4(x, p), p); }
1114 :
1115 : INLINE ulong
1116 750216 : Fl_mul6(ulong x, ulong p)
1117 750216 : { return Fl_sub(Fl_mul8(x, p), Fl_double(x, p), p); }
1118 :
1119 : INLINE ulong
1120 91879 : Fl_mul7(ulong x, ulong p)
1121 91879 : { return Fl_sub(Fl_mul8(x, p), x, p); }
1122 :
1123 : /* Given an elliptic curve E = [a4, a6] over F_p and a nonzero point
1124 : * pt on E, return the quotient E' = E/<P> = [a4_img, a6_img] */
1125 : static void
1126 91888 : Fle_quotient_from_kernel_generator(
1127 : ulong *a4_img, ulong *a6_img, ulong a4, ulong a6, GEN pt, ulong p, ulong pi)
1128 : {
1129 91888 : pari_sp av = avma;
1130 91888 : ulong t = 0, w = 0;
1131 : GEN Q;
1132 : ulong xQ, yQ, tQ, uQ;
1133 :
1134 91888 : Q = gcopy(pt);
1135 : /* Note that, as L is odd, say L = 2n + 1, we necessarily have
1136 : * [(L - 1)/2]P = [n]P = [n - L]P = -[n + 1]P = -[(L + 1)/2]P. This is
1137 : * what the condition Q[1] != xQ tests, so the loop will execute n times. */
1138 : do {
1139 750154 : xQ = uel(Q, 1);
1140 750154 : yQ = uel(Q, 2);
1141 : /* tQ = 6 xQ^2 + b2 xQ + b4
1142 : * = 6 xQ^2 + 2 a4 (since b2 = 0 and b4 = 2 a4) */
1143 750154 : tQ = Fl_add(Fl_mul6(Fl_sqr_pre(xQ, p, pi), p), Fl_double(a4, p), p);
1144 750121 : uQ = Fl_add(Fl_mul4(Fl_sqr_pre(yQ, p, pi), p),
1145 : Fl_mul_pre(tQ, xQ, p, pi), p);
1146 :
1147 750157 : t = Fl_add(t, tQ, p);
1148 750127 : w = Fl_add(w, uQ, p);
1149 750111 : Q = gc_upto(av, Fle_add(pt, Q, a4, p));
1150 750150 : } while (uel(Q, 1) != xQ);
1151 :
1152 91882 : set_avma(av);
1153 : /* a4_img = a4 - 5 * t */
1154 91882 : *a4_img = Fl_sub(a4, Fl_mul5(t, p), p);
1155 : /* a6_img = a6 - b2 * t - 7 * w = a6 - 7 * w (since a1 = a2 = 0 ==> b2 = 0) */
1156 91879 : *a6_img = Fl_sub(a6, Fl_mul7(w, p), p);
1157 91880 : }
1158 :
1159 : /* SECTION: Calculation of modular polynomials. */
1160 :
1161 : /* Given an elliptic curve [a4, a6] over FF_p, try to find a
1162 : * nontrivial L-torsion point on the curve by considering n times a
1163 : * random point; val controls the maximum L-valuation expected of n
1164 : * times a random point */
1165 : static GEN
1166 134373 : find_L_tors_point(
1167 : ulong *ival,
1168 : ulong a4, ulong a6, ulong p, ulong pi,
1169 : ulong n, ulong L, ulong val)
1170 : {
1171 134373 : pari_sp av = avma;
1172 : ulong i;
1173 : GEN P, Q;
1174 : do {
1175 135715 : Q = random_Flj_pre(a4, a6, p, pi);
1176 135708 : P = Flj_mulu_pre(Q, n, a4, p, pi);
1177 135716 : } while (P[3] == 0);
1178 :
1179 260561 : for (i = 0; i < val; ++i) {
1180 218073 : Q = Flj_mulu_pre(P, L, a4, p, pi);
1181 218073 : if (Q[3] == 0) break;
1182 126187 : P = Q;
1183 : }
1184 134374 : if (ival) *ival = i;
1185 134374 : return gc_GEN(av, P);
1186 : }
1187 :
1188 : static GEN
1189 83337 : select_curve_with_L_tors_point(
1190 : ulong *a4, ulong *a6,
1191 : ulong L, ulong j, ulong n, ulong card, ulong val,
1192 : norm_eqn_t ne)
1193 : {
1194 83337 : pari_sp av = avma;
1195 : ulong A4, A4t, A6, A6t;
1196 83337 : ulong p = ne->p, pi = ne->pi;
1197 : GEN P;
1198 83337 : if (card % L != 0) {
1199 0 : pari_err_BUG("select_curve_with_L_tors_point: "
1200 : "Cardinality not divisible by L");
1201 : }
1202 :
1203 83337 : Fl_ellj_to_a4a6(j, p, &A4, &A6);
1204 83336 : Fl_elltwist_disc(A4, A6, ne->T, p, &A4t, &A6t);
1205 :
1206 : /* Either E = [a4, a6] or its twist has cardinality divisible by L
1207 : * because of the choice of p and t earlier on. We find out which
1208 : * by attempting to find a point of order L on each. See bot p16 of
1209 : * Sutherland 2012. */
1210 42490 : while (1) {
1211 : ulong i;
1212 125827 : P = find_L_tors_point(&i, A4, A6, p, pi, n, L, val);
1213 125828 : if (i < val)
1214 83339 : break;
1215 42489 : set_avma(av);
1216 42490 : lswap(A4, A4t);
1217 42490 : lswap(A6, A6t);
1218 : }
1219 83339 : *a4 = A4;
1220 83339 : *a6 = A6; return gc_GEN(av, P);
1221 : }
1222 :
1223 : /* Return 1 if the L-Sylow subgroup of the curve [a4, a6] (mod p) is
1224 : * cyclic, return 0 if it is not cyclic with "high" probability (I
1225 : * guess around 1/L^3 chance it is still cyclic when we return 0).
1226 : *
1227 : * Based on Sutherland's velu.c:velu_verify_Sylow_cyclic() in classpoly-1.0.1 */
1228 : INLINE long
1229 47234 : verify_L_sylow_is_cyclic(long e, ulong a4, ulong a6, ulong p, ulong pi)
1230 : {
1231 : /* Number of times to try to find a point with maximal order in the
1232 : * L-Sylow subgroup. */
1233 : enum { N_RETRIES = 3 };
1234 47234 : pari_sp av = avma;
1235 47234 : long i, res = 0;
1236 : GEN P;
1237 77307 : for (i = 0; i < N_RETRIES; ++i) {
1238 68759 : P = random_Flj_pre(a4, a6, p, pi);
1239 68760 : P = Flj_mulu_pre(P, e, a4, p, pi);
1240 68763 : if (P[3] != 0) { res = 1; break; }
1241 : }
1242 47238 : return gc_long(av,res);
1243 : }
1244 :
1245 : static ulong
1246 83340 : find_noniso_L_isogenous_curve(
1247 : ulong L, ulong n,
1248 : norm_eqn_t ne, long e, ulong val, ulong a4, ulong a6, GEN init_pt, long verify)
1249 : {
1250 : pari_sp ltop, av;
1251 83340 : ulong p = ne->p, pi = ne->pi, j_res = 0;
1252 83340 : GEN pt = init_pt;
1253 83340 : ltop = av = avma;
1254 8546 : while (1) {
1255 : /* c. Use Velu to calculate L-isogenous curve E' = E/<P> */
1256 : ulong a4_img, a6_img;
1257 91886 : ulong z2 = Fl_sqr_pre(pt[3], p, pi);
1258 91888 : pt = mkvecsmall2(Fl_div(pt[1], z2, p),
1259 91886 : Fl_div(pt[2], Fl_mul_pre(z2, pt[3], p, pi), p));
1260 91888 : Fle_quotient_from_kernel_generator(&a4_img, &a6_img,
1261 : a4, a6, pt, p, pi);
1262 :
1263 : /* d. If j(E') = j_res has a different endo ring to j(E), then
1264 : * return j(E'). Otherwise, go to b. */
1265 91880 : if (!verify || verify_L_sylow_is_cyclic(e, a4_img, a6_img, p, pi)) {
1266 83335 : j_res = Fl_ellj_pre(a4_img, a6_img, p, pi);
1267 83342 : break;
1268 : }
1269 :
1270 : /* b. Generate random point P on E of order L */
1271 8546 : set_avma(av);
1272 8546 : pt = find_L_tors_point(NULL, a4, a6, p, pi, n, L, val);
1273 : }
1274 83342 : return gc_ulong(ltop, j_res);
1275 : }
1276 :
1277 : /* Given a prime L and a j-invariant j (mod p), return the j-invariant
1278 : * of a curve which has a different endomorphism ring to j and is
1279 : * L-isogenous to j */
1280 : INLINE ulong
1281 83336 : compute_L_isogenous_curve(
1282 : ulong L, ulong n, norm_eqn_t ne,
1283 : ulong j, ulong card, ulong val, long verify)
1284 : {
1285 : ulong a4, a6;
1286 : long e;
1287 : GEN pt;
1288 :
1289 83336 : if (ne->p < 5 || j == 0 || j == 1728 % ne->p)
1290 0 : pari_err_BUG("compute_L_isogenous_curve");
1291 83336 : pt = select_curve_with_L_tors_point(&a4, &a6, L, j, n, card, val, ne);
1292 83340 : e = card / L;
1293 83340 : if (e * L != card) pari_err_BUG("compute_L_isogenous_curve");
1294 :
1295 83340 : return find_noniso_L_isogenous_curve(L, n, ne, e, val, a4, a6, pt, verify);
1296 : }
1297 :
1298 : INLINE GEN
1299 38689 : get_Lsqr_cycle(const disc_info *dinfo)
1300 : {
1301 38689 : long i, n1 = dinfo->n1, L = dinfo->L;
1302 38689 : GEN cyc = cgetg(L, t_VECSMALL);
1303 38690 : cyc[1] = 0;
1304 315489 : for (i = 2; i <= L / 2; ++i) cyc[i] = cyc[i - 1] + n1;
1305 38690 : if ( ! dinfo->L1) {
1306 123171 : for ( ; i < L; ++i) cyc[i] = cyc[i - 1] + n1;
1307 : } else {
1308 24059 : cyc[L - 1] = 2 * dinfo->n2 - n1 / 2;
1309 206945 : for (i = L - 2; i > L / 2; --i) cyc[i] = cyc[i + 1] - n1;
1310 : }
1311 38690 : return cyc;
1312 : }
1313 :
1314 : INLINE void
1315 534538 : update_Lsqr_cycle(GEN cyc, const disc_info *dinfo)
1316 : {
1317 534538 : long i, L = dinfo->L;
1318 15534514 : for (i = 1; i < L; ++i) ++cyc[i];
1319 534538 : if (dinfo->L1 && cyc[L - 1] == 2 * dinfo->n2) {
1320 22248 : long n1 = dinfo->n1;
1321 198254 : for (i = L / 2 + 1; i < L; ++i) cyc[i] -= n1;
1322 : }
1323 534538 : }
1324 :
1325 : static ulong
1326 38681 : oneroot_of_classpoly(GEN hilb, GEN factu, norm_eqn_t ne, GEN jdb)
1327 : {
1328 38681 : pari_sp av = avma;
1329 38681 : ulong j0, p = ne->p, pi = ne->pi;
1330 38681 : long i, nfactors = lg(gel(factu, 1)) - 1;
1331 38681 : GEN hilbp = ZX_to_Flx(hilb, p);
1332 :
1333 : /* TODO: Work out how to use hilb with better invariant */
1334 38680 : j0 = Flx_oneroot_split_pre(hilbp, p, pi);
1335 38684 : if (j0 == p) {
1336 0 : pari_err_BUG("oneroot_of_classpoly: "
1337 : "Didn't find a root of the class polynomial");
1338 : }
1339 40350 : for (i = 1; i <= nfactors; ++i) {
1340 1663 : long L = gel(factu, 1)[i];
1341 1663 : long val = gel(factu, 2)[i];
1342 1663 : GEN phi = polmodular_db_getp(jdb, L, p);
1343 1666 : val += z_lval(ne->v, L);
1344 1666 : j0 = descend_volcano(phi, j0, p, pi, 0, L, val, val);
1345 1666 : set_avma(av);
1346 : }
1347 38687 : return gc_ulong(av, j0);
1348 : }
1349 :
1350 : /* TODO: Precompute the GEN structs and link them to dinfo */
1351 : INLINE GEN
1352 2880 : make_pcp_surface(const disc_info *dinfo)
1353 : {
1354 2880 : GEN L = mkvecsmall(dinfo->L0);
1355 2880 : GEN n = mkvecsmall(dinfo->n1);
1356 2880 : GEN o = mkvecsmall(dinfo->n1);
1357 2880 : return mkvec2(mkvec3(L, n, o), mkvecsmall3(0, 1, dinfo->n1));
1358 : }
1359 :
1360 : INLINE GEN
1361 2880 : make_pcp_floor(const disc_info *dinfo)
1362 : {
1363 2880 : long k = dinfo->L1 ? 2 : 1;
1364 : GEN L, n, o;
1365 2880 : if (k==1)
1366 : {
1367 1418 : L = mkvecsmall(dinfo->L0);
1368 1418 : n = mkvecsmall(dinfo->n2);
1369 1418 : o = mkvecsmall(dinfo->n2);
1370 : } else
1371 : {
1372 1462 : L = mkvecsmall2(dinfo->L0, dinfo->L1);
1373 1462 : n = mkvecsmall2(dinfo->n2, 2);
1374 1462 : o = mkvecsmall2(dinfo->n2, 2);
1375 : }
1376 2880 : return mkvec2(mkvec3(L, n, o), mkvecsmall3(0, k, dinfo->n2*k));
1377 : }
1378 :
1379 : INLINE GEN
1380 38689 : enum_volcano_surface(norm_eqn_t ne, ulong j0, GEN fdb, GEN G)
1381 : {
1382 38689 : pari_sp av = avma;
1383 38689 : return gc_upto(av, enum_roots(j0, ne, fdb, G, NULL));
1384 : }
1385 :
1386 : INLINE GEN
1387 38688 : enum_volcano_floor(long L, norm_eqn_t ne, ulong j0_pr, GEN fdb, GEN G)
1388 : {
1389 38688 : pari_sp av = avma;
1390 : /* L^2 D is the discriminant for the order R = Z + L OO. */
1391 38688 : long DR = L * L * ne->D;
1392 38688 : long R_cond = L * ne->u; /* conductor(DR); */
1393 38688 : long w = R_cond * ne->v;
1394 : /* TODO: Calculate these once and for all in polmodular0_ZM(). */
1395 : norm_eqn_t eqn;
1396 38688 : memcpy(eqn, ne, sizeof *ne);
1397 38688 : eqn->D = DR;
1398 38688 : eqn->u = R_cond;
1399 38688 : eqn->v = w;
1400 38688 : return gc_upto(av, enum_roots(j0_pr, eqn, fdb, G, NULL));
1401 : }
1402 :
1403 : INLINE void
1404 18667 : carray_reverse_inplace(long *arr, long n)
1405 : {
1406 18667 : long lim = n>>1, i;
1407 18667 : --n;
1408 184421 : for (i = 0; i < lim; i++) lswap(arr[i], arr[n - i]);
1409 18667 : }
1410 :
1411 : INLINE void
1412 573231 : append_neighbours(GEN rts, GEN surface_js, long njs, long L, long m, long i)
1413 : {
1414 573231 : long r_idx = (((i - 1) + m) % njs) + 1; /* (i + m) % njs */
1415 573231 : long l_idx = umodsu((i - 1) - m, njs) + 1; /* (i - m) % njs */
1416 573225 : rts[L] = surface_js[l_idx];
1417 573225 : rts[L + 1] = surface_js[r_idx];
1418 573225 : }
1419 :
1420 : INLINE GEN
1421 41098 : roots_to_coeffs(GEN rts, ulong p, long L)
1422 : {
1423 41098 : long i, k, lrts= lg(rts);
1424 41098 : GEN M = cgetg(L+2+1, t_MAT);
1425 877354 : for (i = 1; i <= L+2; ++i)
1426 836258 : gel(M, i) = cgetg(lrts, t_VECSMALL);
1427 639840 : for (i = 1; i < lrts; ++i) {
1428 598782 : pari_sp av = avma;
1429 598782 : GEN modpol = Flv_roots_to_pol(gel(rts, i), p, 0);
1430 19370132 : for (k = 1; k <= L + 2; ++k) mael(M, k, i) = modpol[k + 1];
1431 598693 : set_avma(av);
1432 : }
1433 41058 : return M;
1434 : }
1435 :
1436 : /* NB: Assumes indices are offset at 0, not at 1 like in GENs;
1437 : * i.e. indices[i] will pick out v[indices[i] + 1] from v. */
1438 : INLINE void
1439 573233 : vecsmall_pick(GEN res, GEN v, GEN indices)
1440 : {
1441 : long i;
1442 16204283 : for (i = 1; i < lg(indices); ++i) res[i] = v[indices[i] + 1];
1443 573233 : }
1444 :
1445 : /* First element of surface_js must lie above the first element of floor_js.
1446 : * Reverse surface_js if it is not oriented in the same direction as floor_js */
1447 : INLINE GEN
1448 38690 : root_matrix(long L, const disc_info *dinfo, long njinvs, GEN surface_js,
1449 : GEN floor_js, ulong n, ulong card, ulong val, norm_eqn_t ne)
1450 : {
1451 : pari_sp av;
1452 38690 : long i, m = dinfo->dl1, njs = lg(surface_js) - 1, inv = dinfo->inv, rev;
1453 38690 : GEN rt_mat = zero_Flm_copy(L + 1, njinvs), rts, cyc;
1454 38689 : ulong p = ne->p, pi = ne->pi, j;
1455 38689 : av = avma;
1456 :
1457 38689 : i = 1;
1458 38689 : cyc = get_Lsqr_cycle(dinfo);
1459 38689 : rts = gel(rt_mat, i);
1460 38689 : vecsmall_pick(rts, floor_js, cyc);
1461 38689 : append_neighbours(rts, surface_js, njs, L, m, i);
1462 :
1463 38689 : i = 2;
1464 38689 : update_Lsqr_cycle(cyc, dinfo);
1465 38689 : rts = gel(rt_mat, i);
1466 38689 : vecsmall_pick(rts, floor_js, cyc);
1467 :
1468 : /* Fix orientation if necessary */
1469 38689 : if (modinv_is_double_eta(inv)) {
1470 : /* TODO: There is potential for refactoring between this,
1471 : * double_eta_initial_js and modfn_preimage. */
1472 5963 : pari_sp av0 = avma;
1473 5963 : GEN F = double_eta_Fl(inv, p);
1474 5963 : pari_sp av = avma;
1475 5963 : ulong r1 = double_eta_power(inv, uel(rts, 1), p, pi);
1476 5963 : GEN r, f = Flx_double_eta_jpoly(F, r1, p, pi);
1477 5963 : if ((j = Flx_oneroot_pre(f, p, pi)) == p) pari_err_BUG("root_matrix");
1478 5963 : j = compute_L_isogenous_curve(L, n, ne, j, card, val, 0);
1479 5963 : set_avma(av);
1480 5963 : r1 = double_eta_power(inv, uel(surface_js, i), p, pi);
1481 5963 : f = Flx_double_eta_jpoly(F, r1, p, pi);
1482 5963 : r = Flx_roots_pre(f, p, pi);
1483 5963 : if (lg(r) != 3) pari_err_BUG("root_matrix");
1484 5963 : rev = (j != uel(r, 1)) && (j != uel(r, 2));
1485 5963 : set_avma(av0);
1486 : } else {
1487 : ulong j1pr, j1;
1488 32726 : j1pr = modfn_preimage(uel(rts, 1), p, pi, dinfo->inv);
1489 32726 : j1 = compute_L_isogenous_curve(L, n, ne, j1pr, card, val, 0);
1490 32726 : rev = j1 != modfn_preimage(uel(surface_js, i), p, pi, dinfo->inv);
1491 : }
1492 38688 : if (rev)
1493 18667 : carray_reverse_inplace(surface_js + 2, njs - 1);
1494 38688 : append_neighbours(rts, surface_js, njs, L, m, i);
1495 :
1496 534550 : for (i = 3; i <= njinvs; ++i) {
1497 495861 : update_Lsqr_cycle(cyc, dinfo);
1498 495867 : rts = gel(rt_mat, i);
1499 495867 : vecsmall_pick(rts, floor_js, cyc);
1500 495865 : append_neighbours(rts, surface_js, njs, L, m, i);
1501 : }
1502 38689 : set_avma(av); return rt_mat;
1503 : }
1504 :
1505 : INLINE void
1506 41442 : interpolate_coeffs(GEN phi_modp, ulong p, GEN j_invs, GEN coeff_mat)
1507 : {
1508 41442 : pari_sp av = avma;
1509 : long i;
1510 41442 : GEN pols = Flv_Flm_polint(j_invs, coeff_mat, p, 0);
1511 879955 : for (i = 1; i < lg(pols); ++i) {
1512 838514 : GEN pol = gel(pols, i);
1513 838514 : long k, maxk = lg(pol);
1514 18362460 : for (k = 2; k < maxk; ++k) coeff(phi_modp, k - 1, i) = pol[k];
1515 : }
1516 41441 : set_avma(av);
1517 41442 : }
1518 :
1519 : INLINE long
1520 345809 : Flv_lastnonzero(GEN v)
1521 : {
1522 : long i;
1523 26794343 : for (i = lg(v) - 1; i > 0; --i)
1524 26793689 : if (v[i]) break;
1525 345809 : return i;
1526 : }
1527 :
1528 : /* Assuming the matrix of coefficients in phi corresponds to polynomials
1529 : * phi_k^* satisfying Y^c phi_k^*(Y^s) for c in {0, 1, ..., s} satisfying
1530 : * c + Lk = L + 1 (mod s), change phi so that the coefficients are for the
1531 : * polynomials Y^c phi_k^*(Y^s) (s is the sparsity factor) */
1532 : INLINE void
1533 10583 : inflate_polys(GEN phi, long L, long s)
1534 : {
1535 10583 : long k, deg = L + 1;
1536 : long maxr;
1537 10583 : maxr = nbrows(phi);
1538 356397 : for (k = 0; k <= deg; ) {
1539 345814 : long i, c = umodsu(L * (1 - k) + 1, s);
1540 : /* TODO: We actually know that the last nonzero element of gel(phi, k)
1541 : * can't be later than index n+1, where n is about (L + 1)/s. */
1542 345811 : ++k;
1543 5477279 : for (i = Flv_lastnonzero(gel(phi, k)); i > 0; --i) {
1544 5131468 : long r = c + (i - 1) * s + 1;
1545 5131468 : if (r > maxr) { coeff(phi, i, k) = 0; continue; }
1546 5058715 : if (r != i) {
1547 4953878 : coeff(phi, r, k) = coeff(phi, i, k);
1548 4953878 : coeff(phi, i, k) = 0;
1549 : }
1550 : }
1551 : }
1552 10583 : }
1553 :
1554 : INLINE void
1555 41381 : Flv_powu_inplace_pre(GEN v, ulong n, ulong p, ulong pi)
1556 : {
1557 : long i;
1558 343546 : for (i = 1; i < lg(v); ++i) v[i] = Fl_powu_pre(v[i], n, p, pi);
1559 41379 : }
1560 :
1561 : INLINE void
1562 10583 : normalise_coeffs(GEN coeffs, GEN js, long L, long s, ulong p, ulong pi)
1563 : {
1564 10583 : pari_sp av = avma;
1565 : long k;
1566 : GEN pows, modinv_js;
1567 :
1568 : /* NB: In fact it would be correct to return the coefficients "as is" when
1569 : * s = 1, but we make that an error anyway since this function should never
1570 : * be called with s = 1. */
1571 10583 : if (s <= 1) pari_err_BUG("normalise_coeffs");
1572 :
1573 : /* pows[i + 1] contains 1 / js[i + 1]^i for i = 0, ..., s - 1. */
1574 10583 : pows = cgetg(s + 1, t_VEC);
1575 10583 : gel(pows, 1) = const_vecsmall(lg(js) - 1, 1);
1576 10583 : modinv_js = Flv_inv_pre(js, p, pi);
1577 10583 : gel(pows, 2) = modinv_js;
1578 39134 : for (k = 3; k <= s; ++k) {
1579 28551 : gel(pows, k) = gcopy(modinv_js);
1580 28551 : Flv_powu_inplace_pre(gel(pows, k), k - 1, p, pi);
1581 : }
1582 :
1583 : /* For each column of coefficients coeffs[k] = [a0 .. an],
1584 : * replace ai by ai / js[i]^c.
1585 : * Said in another way, normalise each row i of coeffs by
1586 : * dividing through by js[i - 1]^c (where c depends on i). */
1587 356459 : for (k = 1; k < lg(coeffs); ++k) {
1588 345814 : long i, c = umodsu(L * (1 - (k - 1)) + 1, s);
1589 345814 : GEN col = gel(coeffs, k), C = gel(pows, c + 1);
1590 5851041 : for (i = 1; i < lg(col); ++i)
1591 5505165 : col[i] = Fl_mul_pre(col[i], C[i], p, pi);
1592 : }
1593 10645 : set_avma(av);
1594 10583 : }
1595 :
1596 : INLINE void
1597 5963 : double_eta_initial_js(
1598 : ulong *x0, ulong *x0pr, ulong j0, ulong j0pr, norm_eqn_t ne,
1599 : long inv, ulong L, ulong n, ulong card, ulong val)
1600 : {
1601 5963 : pari_sp av0 = avma;
1602 5963 : ulong p = ne->p, pi = ne->pi, s2 = ne->s2;
1603 5963 : GEN F = double_eta_Fl(inv, p);
1604 5963 : pari_sp av = avma;
1605 : ulong j1pr, j1, r, t;
1606 : GEN f, g;
1607 :
1608 5963 : *x0pr = modinv_double_eta_from_j(F, inv, j0pr, p, pi, s2);
1609 5963 : t = double_eta_power(inv, *x0pr, p, pi);
1610 5963 : f = Flx_div_by_X_x(Flx_double_eta_jpoly(F, t, p, pi), j0pr, p, &r);
1611 5963 : if (r) pari_err_BUG("double_eta_initial_js");
1612 5963 : j1pr = Flx_deg1_root(f, p);
1613 5963 : set_avma(av);
1614 :
1615 5963 : j1 = compute_L_isogenous_curve(L, n, ne, j1pr, card, val, 0);
1616 5963 : f = Flx_double_eta_xpoly(F, j0, p, pi);
1617 5963 : g = Flx_double_eta_xpoly(F, j1, p, pi);
1618 : /* x0 is the unique common root of f and g */
1619 5963 : *x0 = Flx_deg1_root(Flx_gcd(f, g, p), p);
1620 5963 : set_avma(av0);
1621 :
1622 5963 : if ( ! double_eta_root(inv, x0, *x0, p, pi, s2))
1623 0 : pari_err_BUG("double_eta_initial_js");
1624 5963 : }
1625 :
1626 : /* This is Sutherland 2012, Algorithm 2.1, p16. */
1627 : static GEN
1628 38679 : polmodular_split_p_Flm(ulong L, GEN hilb, GEN factu, norm_eqn_t ne, GEN db,
1629 : GEN G_surface, GEN G_floor, const disc_info *dinfo)
1630 : {
1631 : ulong j0, j0_rt, j0pr, j0pr_rt;
1632 38679 : ulong n, card, val, p = ne->p, pi = ne->pi;
1633 38679 : long inv = dinfo->inv, s = modinv_sparse_factor(inv);
1634 38679 : long nj_selected = ceil((L + 1)/(double)s) + 1;
1635 : GEN surface_js, floor_js, rts, phi_modp, jdb, fdb;
1636 38679 : long switched_signs = 0;
1637 :
1638 38679 : jdb = polmodular_db_for_inv(db, INV_J);
1639 38680 : fdb = polmodular_db_for_inv(db, inv);
1640 :
1641 : /* Precomputation */
1642 38679 : card = p + 1 - ne->t;
1643 38679 : val = u_lvalrem(card, L, &n); /* n = card / L^{v_L(card)} */
1644 :
1645 38681 : j0 = oneroot_of_classpoly(hilb, factu, ne, jdb);
1646 38687 : j0pr = compute_L_isogenous_curve(L, n, ne, j0, card, val, 1);
1647 38689 : if (modinv_is_double_eta(inv)) {
1648 5963 : double_eta_initial_js(&j0_rt, &j0pr_rt, j0, j0pr, ne, inv, L, n, card, val);
1649 : } else {
1650 32726 : j0_rt = modfn_root(j0, ne, inv);
1651 32726 : j0pr_rt = modfn_root(j0pr, ne, inv);
1652 : }
1653 38689 : surface_js = enum_volcano_surface(ne, j0_rt, fdb, G_surface);
1654 38688 : floor_js = enum_volcano_floor(L, ne, j0pr_rt, fdb, G_floor);
1655 38690 : rts = root_matrix(L, dinfo, nj_selected, surface_js, floor_js,
1656 : n, card, val, ne);
1657 2409 : do {
1658 41098 : pari_sp btop = avma;
1659 : long i;
1660 : GEN coeffs, surf;
1661 :
1662 41098 : coeffs = roots_to_coeffs(rts, p, L);
1663 41097 : surf = vecsmall_shorten(surface_js, nj_selected);
1664 41097 : if (s > 1) {
1665 10583 : normalise_coeffs(coeffs, surf, L, s, p, pi);
1666 10583 : Flv_powu_inplace_pre(surf, s, p, pi);
1667 : }
1668 41097 : phi_modp = zero_Flm_copy(L + 2, L + 2);
1669 41099 : interpolate_coeffs(phi_modp, p, surf, coeffs);
1670 41099 : if (s > 1) inflate_polys(phi_modp, L, s);
1671 :
1672 : /* TODO: Calculate just this coefficient of X^L Y^L, so we can do this
1673 : * test, then calculate the other coefficients; at the moment we are
1674 : * sometimes doing all the roots-to-coeffs, normalisation and interpolation
1675 : * work twice. */
1676 41099 : if (ucoeff(phi_modp, L + 1, L + 1) == p - 1) break;
1677 :
1678 2409 : if (switched_signs) pari_err_BUG("polmodular_split_p_Flm");
1679 :
1680 2409 : set_avma(btop);
1681 28120 : for (i = 1; i < lg(rts); ++i) {
1682 25711 : surface_js[i] = Fl_neg(surface_js[i], p);
1683 25711 : coeff(rts, L, i) = Fl_neg(coeff(rts, L, i), p);
1684 25711 : coeff(rts, L + 1, i) = Fl_neg(coeff(rts, L + 1, i), p);
1685 : }
1686 2409 : switched_signs = 1;
1687 : } while (1);
1688 38690 : dbg_printf(4)(" Phi_%lu(X, Y) (mod %lu) = %Ps\n", L, p, phi_modp);
1689 :
1690 38690 : return phi_modp;
1691 : }
1692 :
1693 : INLINE void
1694 2464 : Flv_deriv_pre_inplace(GEN v, long deg, ulong p, ulong pi)
1695 : {
1696 2464 : long i, ln = lg(v), d = deg % p;
1697 57207 : for (i = ln - 1; i > 1; --i, --d) v[i] = Fl_mul_pre(v[i - 1], d, p, pi);
1698 2464 : v[1] = 0;
1699 2464 : }
1700 :
1701 : INLINE GEN
1702 2674 : eval_modpoly_modp(GEN Tp, GEN j_powers, ulong p, ulong pi, int compute_derivs)
1703 : {
1704 2674 : long L = lg(j_powers) - 3;
1705 2674 : GEN j_pows_p = ZV_to_Flv(j_powers, p);
1706 2673 : GEN tmp = cgetg(2 + 2 * compute_derivs, t_VEC);
1707 : /* We wrap the result in this t_VEC Tp to trick the
1708 : * ZM_*_CRT() functions into thinking it's a matrix. */
1709 2673 : gel(tmp, 1) = Flm_Flc_mul_pre(Tp, j_pows_p, p, pi);
1710 2674 : if (compute_derivs) {
1711 1232 : Flv_deriv_pre_inplace(j_pows_p, L + 1, p, pi);
1712 1232 : gel(tmp, 2) = Flm_Flc_mul_pre(Tp, j_pows_p, p, pi);
1713 1232 : Flv_deriv_pre_inplace(j_pows_p, L + 1, p, pi);
1714 1232 : gel(tmp, 3) = Flm_Flc_mul_pre(Tp, j_pows_p, p, pi);
1715 : }
1716 2674 : return tmp;
1717 : }
1718 :
1719 : /* Parallel interface */
1720 : GEN
1721 38684 : polmodular_worker(GEN tp, ulong L, GEN hilb, GEN factu, GEN vne, GEN vinfo,
1722 : long derivs, GEN j_powers, GEN G_surface, GEN G_floor,
1723 : GEN fdb)
1724 : {
1725 38684 : pari_sp av = avma;
1726 : norm_eqn_t ne;
1727 38684 : long D = vne[1], u = vne[2];
1728 38684 : ulong vL, t = tp[1], p = tp[2];
1729 : GEN Tp;
1730 :
1731 38684 : if (! uissquareall((4 * p - t * t) / -D, &vL))
1732 0 : pari_err_BUG("polmodular_worker");
1733 38687 : norm_eqn_set(ne, D, t, u, vL, NULL, p); /* L | vL */
1734 38678 : Tp = polmodular_split_p_Flm(L, hilb, factu, ne, fdb,
1735 : G_surface, G_floor, (const disc_info*)vinfo);
1736 38690 : if (!isintzero(j_powers))
1737 2674 : Tp = eval_modpoly_modp(Tp, j_powers, ne->p, ne->pi, derivs);
1738 38690 : return gc_upto(av, Tp);
1739 : }
1740 :
1741 : static GEN
1742 24701 : sympol_to_ZM(GEN phi, long L)
1743 : {
1744 24701 : pari_sp av = avma;
1745 24701 : GEN res = zeromatcopy(L + 2, L + 2);
1746 24701 : long i, j, c = 1;
1747 108041 : for (i = 1; i <= L + 1; ++i)
1748 276080 : for (j = 1; j <= i; ++j, ++c)
1749 192740 : gcoeff(res, i, j) = gcoeff(res, j, i) = gel(phi, c);
1750 24701 : gcoeff(res, L + 2, 1) = gcoeff(res, 1, L + 2) = gen_1;
1751 24701 : return gc_GEN(av, res);
1752 : }
1753 :
1754 : static GEN polmodular_small_ZM(long L, long inv, GEN *db);
1755 :
1756 : INLINE long
1757 27835 : modinv_max_internal_level(long inv)
1758 : {
1759 27835 : switch (inv) {
1760 25235 : case INV_J: return 5;
1761 322 : case INV_G2: return 2;
1762 429 : case INV_F:
1763 : case INV_F2:
1764 : case INV_F4:
1765 429 : case INV_F8: return 5;
1766 224 : case INV_W2W5:
1767 224 : case INV_W2W5E2: return 7;
1768 483 : case INV_W2W3:
1769 : case INV_W2W3E2:
1770 : case INV_W3W3:
1771 483 : case INV_W3W7: return 5;
1772 63 : case INV_W3W3E2:return 2;
1773 722 : case INV_F3:
1774 : case INV_W2W7:
1775 : case INV_W2W7E2:
1776 722 : case INV_W2W13: return 3;
1777 357 : case INV_W3W5:
1778 : case INV_W5W7:
1779 : case INV_W3W13:
1780 357 : case INV_ATKIN3: return 2;
1781 : }
1782 : pari_err_BUG("modinv_max_internal_level"); return LONG_MAX;/*LCOV_EXCL_LINE*/
1783 : }
1784 : static void
1785 45 : db_add_levels(GEN *db, GEN P, long inv)
1786 45 : { polmodular_db_add_levels(db, zv_to_longptr(P), lg(P)-1, inv); }
1787 :
1788 : GEN
1789 27716 : polmodular0_ZM(long L, long inv, GEN J, GEN Q, int compute_derivs, GEN *db)
1790 : {
1791 27716 : pari_sp ltop = avma;
1792 27716 : long k, d, Dcnt, nprimes = 0;
1793 : GEN modpoly, plist, tp, j_powers;
1794 : disc_info Ds[MODPOLY_MAX_DCNT];
1795 27716 : long lvl = modinv_level(inv);
1796 27716 : if (ugcd(L, lvl) != 1)
1797 7 : pari_err_DOMAIN("polmodular0_ZM", "invariant",
1798 : "incompatible with", stoi(L), stoi(lvl));
1799 :
1800 27709 : dbg_printf(1)("Calculating modular polynomial of level %lu for invariant %d\n", L, inv);
1801 27709 : if (L <= modinv_max_internal_level(inv)) return polmodular_small_ZM(L,inv,db);
1802 :
1803 2861 : Dcnt = discriminant_with_classno_at_least(Ds, L, inv, Q, USE_SPARSE_FACTOR);
1804 5741 : for (d = 0; d < Dcnt; d++) nprimes += Ds[d].nprimes;
1805 2861 : modpoly = cgetg(nprimes+1, t_VEC);
1806 2861 : plist = cgetg(nprimes+1, t_VECSMALL);
1807 2861 : tp = mkvec(mkvecsmall2(0,0));
1808 2861 : j_powers = gen_0;
1809 2861 : if (J) {
1810 63 : compute_derivs = !!compute_derivs;
1811 63 : j_powers = Fp_powers(J, L+1, Q);
1812 : }
1813 5741 : for (d = 0, k = 1; d < Dcnt; d++)
1814 : {
1815 2880 : disc_info *dinfo = &Ds[d];
1816 : struct pari_mt pt;
1817 2880 : const long D = dinfo->D1, DK = dinfo->D0;
1818 2880 : const ulong cond = usqrt(D / DK);
1819 2880 : long i, pending = 0;
1820 2880 : GEN worker, hilb, factu = factoru(cond);
1821 :
1822 2880 : polmodular_db_add_level(db, dinfo->L0, inv);
1823 2880 : if (dinfo->L1) polmodular_db_add_level(db, dinfo->L1, inv);
1824 2880 : dbg_printf(1)("Selected discriminant D = %ld = %ld^2 * %ld.\n", D,cond,DK);
1825 2880 : hilb = polclass0(DK, INV_J, 0, db);
1826 2880 : if (cond > 1) db_add_levels(db, gel(factu,1), INV_J);
1827 2880 : dbg_printf(1)("D = %ld, L0 = %lu, L1 = %lu, ", dinfo->D1, dinfo->L0, dinfo->L1);
1828 2880 : dbg_printf(1)("n1 = %lu, n2 = %lu, dl1 = %lu, dl2_0 = %lu, dl2_1 = %lu\n",
1829 : dinfo->n1, dinfo->n2, dinfo->dl1, dinfo->dl2_0, dinfo->dl2_1);
1830 2880 : dbg_printf(0)("Calculating modular polynomial of level %lu:", L);
1831 :
1832 2880 : worker = snm_closure(is_entry("_polmodular_worker"),
1833 : mkvecn(10, utoi(L), hilb, factu, mkvecsmall2(D, cond),
1834 : (GEN)dinfo, stoi(compute_derivs), j_powers,
1835 : make_pcp_surface(dinfo),
1836 : make_pcp_floor(dinfo), *db));
1837 2880 : mt_queue_start_lim(&pt, worker, dinfo->nprimes);
1838 45586 : for (i = 0; i < dinfo->nprimes || pending; i++)
1839 : {
1840 : long workid;
1841 : GEN done;
1842 42706 : if (i < dinfo->nprimes)
1843 : {
1844 38690 : mael(tp, 1, 1) = dinfo->traces[i];
1845 38690 : mael(tp, 1, 2) = dinfo->primes[i];
1846 : }
1847 42706 : mt_queue_submit(&pt, i, i < dinfo->nprimes? tp: NULL);
1848 42706 : done = mt_queue_get(&pt, &workid, &pending);
1849 42706 : if (done)
1850 : {
1851 38690 : plist[k] = dinfo->primes[workid];
1852 38690 : gel(modpoly, k) = done; k++;
1853 38690 : dbg_printf(0)(" %ld%%", k*100/nprimes);
1854 : }
1855 : }
1856 2880 : dbg_printf(0)(" done\n");
1857 2880 : mt_queue_end(&pt);
1858 2880 : killblock((GEN)dinfo->primes);
1859 : }
1860 2861 : modpoly = nmV_chinese_center(modpoly, plist, NULL);
1861 2861 : if (J) modpoly = FpM_red(modpoly, Q);
1862 2861 : return gc_upto(ltop, modpoly);
1863 : }
1864 :
1865 : GEN
1866 19245 : polmodular_ZM(long L, long inv)
1867 : {
1868 : GEN db, Phi;
1869 :
1870 19245 : if (L < 2)
1871 7 : pari_err_DOMAIN("polmodular_ZM", "L", "<", gen_2, stoi(L));
1872 :
1873 : /* TODO: Handle nonprime L. Algorithm 1.1 and Corollary 3.4 in Sutherland,
1874 : * "Class polynomials for nonholomorphic modular functions" */
1875 19238 : if (! uisprime(L)) pari_err_IMPL("composite level");
1876 :
1877 19231 : db = polmodular_db_init(inv);
1878 19231 : Phi = polmodular0_ZM(L, inv, NULL, NULL, 0, &db);
1879 19224 : gunclone_deep(db); return Phi;
1880 : }
1881 :
1882 : GEN
1883 19161 : polmodular_ZXX(long L, long inv, long vx, long vy)
1884 : {
1885 19161 : pari_sp av = avma;
1886 19161 : GEN phi = polmodular_ZM(L, inv);
1887 :
1888 19140 : if (vx < 0) vx = 0;
1889 19140 : if (vy < 0) vy = 1;
1890 19140 : if (varncmp(vx, vy) >= 0)
1891 14 : pari_err_PRIORITY("polmodular_ZXX", pol_x(vx), "<=", vy);
1892 19126 : return gc_GEN(av, RgM_to_RgXX(phi, vx, vy));
1893 : }
1894 :
1895 : INLINE GEN
1896 56 : FpV_deriv(GEN v, long deg, GEN P)
1897 : {
1898 56 : long i, ln = lg(v);
1899 56 : GEN dv = cgetg(ln, t_VEC);
1900 392 : for (i = ln-1; i > 1; i--, deg--) gel(dv, i) = Fp_mulu(gel(v, i-1), deg, P);
1901 56 : gel(dv, 1) = gen_0; return dv;
1902 : }
1903 :
1904 : GEN
1905 126 : Fp_polmodular_evalx(long L, long inv, GEN J, GEN P, long v, int compute_derivs)
1906 : {
1907 126 : pari_sp av = avma;
1908 : GEN db, phi;
1909 :
1910 126 : if (L <= modinv_max_internal_level(inv)) {
1911 : GEN tmp;
1912 63 : GEN phi = RgM_to_FpM(polmodular_ZM(L, inv), P);
1913 63 : GEN j_powers = Fp_powers(J, L + 1, P);
1914 63 : GEN modpol = RgV_to_RgX(FpM_FpC_mul(phi, j_powers, P), v);
1915 63 : if (compute_derivs) {
1916 28 : tmp = cgetg(4, t_VEC);
1917 28 : gel(tmp, 1) = modpol;
1918 28 : j_powers = FpV_deriv(j_powers, L + 1, P);
1919 28 : gel(tmp, 2) = RgV_to_RgX(FpM_FpC_mul(phi, j_powers, P), v);
1920 28 : j_powers = FpV_deriv(j_powers, L + 1, P);
1921 28 : gel(tmp, 3) = RgV_to_RgX(FpM_FpC_mul(phi, j_powers, P), v);
1922 : } else
1923 35 : tmp = modpol;
1924 63 : return gc_GEN(av, tmp);
1925 : }
1926 :
1927 63 : db = polmodular_db_init(inv);
1928 63 : phi = polmodular0_ZM(L, inv, J, P, compute_derivs, &db);
1929 63 : phi = RgM_to_RgXV(phi, v);
1930 63 : gunclone_deep(db);
1931 63 : return gc_GEN(av, compute_derivs? phi: gel(phi, 1));
1932 : }
1933 :
1934 : GEN
1935 630 : polmodular(long L, long inv, GEN x, long v, long compute_derivs)
1936 : {
1937 630 : pari_sp av = avma;
1938 : long tx;
1939 630 : GEN J = NULL, P = NULL, res = NULL, one = NULL;
1940 :
1941 630 : check_modinv(inv);
1942 623 : if (!x || gequalX(x)) {
1943 483 : long xv = 0;
1944 483 : if (x) xv = varn(x);
1945 483 : if (compute_derivs) pari_err_FLAG("polmodular");
1946 476 : return polmodular_ZXX(L, inv, xv, v);
1947 : }
1948 :
1949 140 : tx = typ(x);
1950 140 : if (tx == t_INTMOD) {
1951 63 : J = gel(x, 2);
1952 63 : P = gel(x, 1);
1953 63 : one = mkintmod(gen_1, P);
1954 77 : } else if (tx == t_FFELT) {
1955 70 : J = FF_to_FpXQ_i(x);
1956 70 : if (degpol(J) > 0)
1957 7 : pari_err_DOMAIN("polmodular", "x", "not in prime subfield ", gen_0, x);
1958 63 : J = constant_coeff(J);
1959 63 : P = FF_p_i(x);
1960 63 : one = FF_1(x);
1961 : } else
1962 7 : pari_err_TYPE("polmodular", x);
1963 :
1964 126 : if (v < 0) v = 1;
1965 126 : res = Fp_polmodular_evalx(L, inv, J, P, v, compute_derivs);
1966 126 : return gc_upto(av, gmul(res, one));
1967 : }
1968 :
1969 : /* SECTION: Modular polynomials of level <= MAX_INTERNAL_MODPOLY_LEVEL. */
1970 :
1971 : /* These functions return a vector of coefficients of classical modular
1972 : * polynomials Phi_L(X,Y) of small level L. The number of such coefficients is
1973 : * (L+1)(L+2)/2 since Phi is symmetric. We omit the common coefficient of
1974 : * X^{L+1} and Y^{L+1} since it is always 1. Use sympol_to_ZM() to get the
1975 : * corresponding desymmetrised matrix of coefficients */
1976 :
1977 : /* Phi2, the modular polynomial of level 2:
1978 : *
1979 : * X^3 + X^2 * (-Y^2 + 1488*Y - 162000)
1980 : * + X * (1488*Y^2 + 40773375*Y + 8748000000)
1981 : * + Y^3 - 162000*Y^2 + 8748000000*Y - 157464000000000
1982 : *
1983 : * [[3, 0, 1],
1984 : * [2, 2, -1],
1985 : * [2, 1, 1488],
1986 : * [2, 0, -162000],
1987 : * [1, 1, 40773375],
1988 : * [1, 0, 8748000000],
1989 : * [0, 0, -157464000000000]], */
1990 : static GEN
1991 19994 : phi2_ZV(void)
1992 : {
1993 19994 : GEN phi2 = cgetg(7, t_VEC);
1994 19994 : gel(phi2, 1) = uu32toi(36662, 1908994048);
1995 19994 : setsigne(gel(phi2, 1), -1);
1996 19994 : gel(phi2, 2) = uu32toi(2, 158065408);
1997 19994 : gel(phi2, 3) = stoi(40773375);
1998 19994 : gel(phi2, 4) = stoi(-162000);
1999 19994 : gel(phi2, 5) = stoi(1488);
2000 19994 : gel(phi2, 6) = gen_m1;
2001 19994 : return phi2;
2002 : }
2003 :
2004 : /* L = 3
2005 : *
2006 : * [4, 0, 1],
2007 : * [3, 3, -1],
2008 : * [3, 2, 2232],
2009 : * [3, 1, -1069956],
2010 : * [3, 0, 36864000],
2011 : * [2, 2, 2587918086],
2012 : * [2, 1, 8900222976000],
2013 : * [2, 0, 452984832000000],
2014 : * [1, 1, -770845966336000000],
2015 : * [1, 0, 1855425871872000000000]
2016 : * [0, 0, 0]
2017 : *
2018 : * 1855425871872000000000 = 2^32 * (100 * 2^32 + 2503270400) */
2019 : static GEN
2020 1889 : phi3_ZV(void)
2021 : {
2022 1889 : GEN phi3 = cgetg(11, t_VEC);
2023 1889 : pari_sp av = avma;
2024 1889 : gel(phi3, 1) = gen_0;
2025 1889 : gel(phi3, 2) = gc_upto(av, shifti(uu32toi(100, 2503270400UL), 32));
2026 1889 : gel(phi3, 3) = uu32toi(179476562, 2147483648UL);
2027 1889 : setsigne(gel(phi3, 3), -1);
2028 1889 : gel(phi3, 4) = uu32toi(105468, 3221225472UL);
2029 1889 : gel(phi3, 5) = uu32toi(2072, 1050738688);
2030 1889 : gel(phi3, 6) = utoi(2587918086UL);
2031 1889 : gel(phi3, 7) = stoi(36864000);
2032 1889 : gel(phi3, 8) = stoi(-1069956);
2033 1889 : gel(phi3, 9) = stoi(2232);
2034 1889 : gel(phi3, 10) = gen_m1;
2035 1889 : return phi3;
2036 : }
2037 :
2038 : static GEN
2039 1859 : phi5_ZV(void)
2040 : {
2041 1859 : GEN phi5 = cgetg(22, t_VEC);
2042 1859 : gel(phi5, 1) = mkintn(5, 0x18c2cc9cUL, 0x484382b2UL, 0xdc000000UL, 0x0UL, 0x0UL);
2043 1859 : gel(phi5, 2) = mkintn(5, 0x2638fUL, 0x2ff02690UL, 0x68026000UL, 0x0UL, 0x0UL);
2044 1859 : gel(phi5, 3) = mkintn(5, 0x308UL, 0xac9d9a4UL, 0xe0fdab12UL, 0xc0000000UL, 0x0UL);
2045 1859 : setsigne(gel(phi5, 3), -1);
2046 1859 : gel(phi5, 4) = mkintn(5, 0x13UL, 0xaae09f9dUL, 0x1b5ef872UL, 0x30000000UL, 0x0UL);
2047 1859 : gel(phi5, 5) = mkintn(4, 0x1b802fa9UL, 0x77ba0653UL, 0xd2f78000UL, 0x0UL);
2048 1859 : gel(phi5, 6) = mkintn(4, 0xfbfdUL, 0x278e4756UL, 0xdf08a7c4UL, 0x40000000UL);
2049 1859 : gel(phi5, 7) = mkintn(4, 0x35f922UL, 0x62ccea6fUL, 0x153d0000UL, 0x0UL);
2050 1859 : gel(phi5, 8) = mkintn(4, 0x97dUL, 0x29203fafUL, 0xc3036909UL, 0x80000000UL);
2051 1859 : setsigne(gel(phi5, 8), -1);
2052 1859 : gel(phi5, 9) = mkintn(3, 0x56e9e892UL, 0xd7781867UL, 0xf2ea0000UL);
2053 1859 : gel(phi5, 10) = mkintn(3, 0x5d6dUL, 0xe0a58f4eUL, 0x9ee68c14UL);
2054 1859 : setsigne(gel(phi5, 10), -1);
2055 1859 : gel(phi5, 11) = mkintn(3, 0x1100dUL, 0x85cea769UL, 0x40000000UL);
2056 1859 : gel(phi5, 12) = mkintn(3, 0x1b38UL, 0x43cf461fUL, 0x3a900000UL);
2057 1859 : gel(phi5, 13) = mkintn(3, 0x14UL, 0xc45a616eUL, 0x4801680fUL);
2058 1859 : gel(phi5, 14) = uu32toi(0x17f4350UL, 0x493ca3e0UL);
2059 1859 : gel(phi5, 15) = uu32toi(0x183UL, 0xe54ce1f8UL);
2060 1859 : gel(phi5, 16) = uu32toi(0x1c9UL, 0x18860000UL);
2061 1859 : gel(phi5, 17) = uu32toi(0x39UL, 0x6f7a2206UL);
2062 1859 : setsigne(gel(phi5, 17), -1);
2063 1859 : gel(phi5, 18) = stoi(2028551200);
2064 1859 : gel(phi5, 19) = stoi(-4550940);
2065 1859 : gel(phi5, 20) = stoi(3720);
2066 1859 : gel(phi5, 21) = gen_m1;
2067 1859 : return phi5;
2068 : }
2069 :
2070 : static GEN
2071 182 : phi5_f_ZV(void)
2072 : {
2073 182 : GEN phi = zerovec(21);
2074 182 : gel(phi, 3) = stoi(4);
2075 182 : gel(phi, 21) = gen_m1;
2076 182 : return phi;
2077 : }
2078 :
2079 : static GEN
2080 21 : phi3_f3_ZV(void)
2081 : {
2082 21 : GEN phi = zerovec(10);
2083 21 : gel(phi, 3) = stoi(8);
2084 21 : gel(phi, 10) = gen_m1;
2085 21 : return phi;
2086 : }
2087 :
2088 : static GEN
2089 119 : phi2_g2_ZV(void)
2090 : {
2091 119 : GEN phi = zerovec(6);
2092 119 : gel(phi, 1) = stoi(-54000);
2093 119 : gel(phi, 3) = stoi(495);
2094 119 : gel(phi, 6) = gen_m1;
2095 119 : return phi;
2096 : }
2097 :
2098 : static GEN
2099 56 : phi5_w2w3_ZV(void)
2100 : {
2101 56 : GEN phi = zerovec(21);
2102 56 : gel(phi, 3) = gen_m1;
2103 56 : gel(phi, 10) = stoi(5);
2104 56 : gel(phi, 21) = gen_m1;
2105 56 : return phi;
2106 : }
2107 :
2108 : static GEN
2109 98 : phi7_w2w5_ZV(void)
2110 : {
2111 98 : GEN phi = zerovec(36);
2112 98 : gel(phi, 3) = gen_m1;
2113 98 : gel(phi, 15) = stoi(56);
2114 98 : gel(phi, 19) = stoi(42);
2115 98 : gel(phi, 24) = stoi(21);
2116 98 : gel(phi, 30) = stoi(7);
2117 98 : gel(phi, 36) = gen_m1;
2118 98 : return phi;
2119 : }
2120 :
2121 : static GEN
2122 63 : phi5_w3w3_ZV(void)
2123 : {
2124 63 : GEN phi = zerovec(21);
2125 63 : gel(phi, 3) = stoi(9);
2126 63 : gel(phi, 6) = stoi(-15);
2127 63 : gel(phi, 15) = stoi(5);
2128 63 : gel(phi, 21) = gen_m1;
2129 63 : return phi;
2130 : }
2131 :
2132 : static GEN
2133 196 : phi3_w2w7_ZV(void)
2134 : {
2135 196 : GEN phi = zerovec(10);
2136 196 : gel(phi, 3) = gen_m1;
2137 196 : gel(phi, 6) = stoi(3);
2138 196 : gel(phi, 10) = gen_m1;
2139 196 : return phi;
2140 : }
2141 :
2142 : static GEN
2143 35 : phi2_w3w5_ZV(void)
2144 : {
2145 35 : GEN phi = zerovec(6);
2146 35 : gel(phi, 3) = gen_1;
2147 35 : gel(phi, 6) = gen_m1;
2148 35 : return phi;
2149 : }
2150 :
2151 : static GEN
2152 42 : phi5_w3w7_ZV(void)
2153 : {
2154 42 : GEN phi = zerovec(21);
2155 42 : gel(phi, 3) = gen_m1;
2156 42 : gel(phi, 6) = stoi(10);
2157 42 : gel(phi, 8) = stoi(5);
2158 42 : gel(phi, 10) = stoi(35);
2159 42 : gel(phi, 13) = stoi(20);
2160 42 : gel(phi, 15) = stoi(10);
2161 42 : gel(phi, 17) = stoi(5);
2162 42 : gel(phi, 19) = stoi(5);
2163 42 : gel(phi, 21) = gen_m1;
2164 42 : return phi;
2165 : }
2166 :
2167 : static GEN
2168 35 : phi3_w2w13_ZV(void)
2169 : {
2170 35 : GEN phi = zerovec(10);
2171 35 : gel(phi, 3) = gen_m1;
2172 35 : gel(phi, 6) = stoi(3);
2173 35 : gel(phi, 8) = stoi(3);
2174 35 : gel(phi, 10) = gen_m1;
2175 35 : return phi;
2176 : }
2177 :
2178 : static GEN
2179 21 : phi2_w3w3e2_ZV(void)
2180 : {
2181 21 : GEN phi = zerovec(6);
2182 21 : gel(phi, 3) = stoi(3);
2183 21 : gel(phi, 6) = gen_m1;
2184 21 : return phi;
2185 : }
2186 :
2187 : static GEN
2188 56 : phi2_w5w7_ZV(void)
2189 : {
2190 56 : GEN phi = zerovec(6);
2191 56 : gel(phi, 3) = gen_1;
2192 56 : gel(phi, 5) = gen_2;
2193 56 : gel(phi, 6) = gen_m1;
2194 56 : return phi;
2195 : }
2196 :
2197 : static GEN
2198 14 : phi2_w3w13_ZV(void)
2199 : {
2200 14 : GEN phi = zerovec(6);
2201 14 : gel(phi, 3) = gen_m1;
2202 14 : gel(phi, 5) = gen_2;
2203 14 : gel(phi, 6) = gen_m1;
2204 14 : return phi;
2205 : }
2206 :
2207 : static GEN
2208 21 : phi2_atkin3_ZV(void)
2209 : {
2210 21 : GEN phi = zerovec(6);
2211 21 : gel(phi, 1) = utoi(28166076);
2212 21 : gel(phi, 2) = utoi(741474);
2213 21 : gel(phi, 3) = utoi(17343);
2214 21 : gel(phi, 4) = utoi(1566);
2215 21 : gel(phi, 6) = gen_m1;
2216 21 : return phi;
2217 : }
2218 :
2219 : INLINE long
2220 147 : modinv_parent(long inv)
2221 : {
2222 147 : switch (inv) {
2223 42 : case INV_F2:
2224 : case INV_F4:
2225 42 : case INV_F8: return INV_F;
2226 14 : case INV_W2W3E2: return INV_W2W3;
2227 21 : case INV_W2W5E2: return INV_W2W5;
2228 70 : case INV_W2W7E2: return INV_W2W7;
2229 0 : case INV_W3W3E2: return INV_W3W3;
2230 : default: pari_err_BUG("modinv_parent"); return -1;/*LCOV_EXCL_LINE*/
2231 : }
2232 : }
2233 :
2234 : /* TODO: Think of a better name than "parent power"; sheesh. */
2235 : INLINE long
2236 147 : modinv_parent_power(long inv)
2237 : {
2238 147 : switch (inv) {
2239 14 : case INV_F4: return 4;
2240 14 : case INV_F8: return 8;
2241 119 : case INV_F2:
2242 : case INV_W2W3E2:
2243 : case INV_W2W5E2:
2244 : case INV_W2W7E2:
2245 119 : case INV_W3W3E2: return 2;
2246 : default: pari_err_BUG("modinv_parent_power"); return -1;/*LCOV_EXCL_LINE*/
2247 : }
2248 : }
2249 :
2250 : static GEN
2251 147 : polmodular0_powerup_ZM(long L, long inv, GEN *db)
2252 : {
2253 147 : pari_sp ltop = avma, av;
2254 : long s, D, nprimes, N;
2255 : GEN mp, pol, P, H;
2256 147 : long parent = modinv_parent(inv);
2257 147 : long e = modinv_parent_power(inv);
2258 : disc_info Ds[MODPOLY_MAX_DCNT];
2259 : /* FIXME: We throw away the table of fundamental discriminants here. */
2260 147 : long nDs = discriminant_with_classno_at_least(Ds, L, inv, NULL, IGNORE_SPARSE_FACTOR);
2261 147 : if (nDs != 1) pari_err_BUG("polmodular0_powerup_ZM");
2262 147 : D = Ds[0].D1;
2263 147 : nprimes = Ds[0].nprimes + 1;
2264 147 : mp = polmodular0_ZM(L, parent, NULL, NULL, 0, db);
2265 147 : H = polclass0(D, parent, 0, db);
2266 :
2267 147 : N = L + 2;
2268 147 : if (degpol(H) < N) pari_err_BUG("polmodular0_powerup_ZM");
2269 :
2270 147 : av = avma;
2271 147 : pol = ZM_init_CRT(zero_Flm_copy(N, L + 2), 1);
2272 147 : P = gen_1;
2273 490 : for (s = 1; s < nprimes; ++s) {
2274 : pari_sp av1, av2;
2275 343 : ulong p = Ds[0].primes[s-1], pi = get_Fl_red(p);
2276 : long i;
2277 : GEN Hrts, js, Hp, Phip, coeff_mat, phi_modp;
2278 :
2279 343 : phi_modp = zero_Flm_copy(N, L + 2);
2280 343 : av1 = avma;
2281 343 : Hp = ZX_to_Flx(H, p);
2282 343 : Hrts = Flx_roots_pre(Hp, p, pi);
2283 343 : if (lg(Hrts)-1 < N) pari_err_BUG("polmodular0_powerup_ZM");
2284 343 : js = cgetg(N + 1, t_VECSMALL);
2285 2590 : for (i = 1; i <= N; ++i)
2286 2247 : uel(js, i) = Fl_powu_pre(uel(Hrts, i), e, p, pi);
2287 :
2288 343 : Phip = ZM_to_Flm(mp, p);
2289 343 : coeff_mat = zero_Flm_copy(N, L + 2);
2290 343 : av2 = avma;
2291 2590 : for (i = 1; i <= N; ++i) {
2292 : long k;
2293 : GEN phi_at_ji, mprts;
2294 :
2295 2247 : phi_at_ji = Flm_Fl_polmodular_evalx(Phip, L, uel(Hrts, i), p, pi);
2296 2247 : mprts = Flx_roots_pre(phi_at_ji, p, pi);
2297 2247 : if (lg(mprts) != L + 2) pari_err_BUG("polmodular0_powerup_ZM");
2298 :
2299 2247 : Flv_powu_inplace_pre(mprts, e, p, pi);
2300 2247 : phi_at_ji = Flv_roots_to_pol(mprts, p, 0);
2301 :
2302 17710 : for (k = 1; k <= L + 2; ++k)
2303 15463 : ucoeff(coeff_mat, i, k) = uel(phi_at_ji, k + 1);
2304 2247 : set_avma(av2);
2305 : }
2306 :
2307 343 : interpolate_coeffs(phi_modp, p, js, coeff_mat);
2308 343 : set_avma(av1);
2309 :
2310 343 : (void) ZM_incremental_CRT(&pol, phi_modp, &P, p);
2311 343 : if (gc_needed(av, 2)) (void)gc_all(av, 2, &pol, &P);
2312 : }
2313 147 : killblock((GEN)Ds[0].primes); return gc_upto(ltop, pol);
2314 : }
2315 :
2316 : /* Returns the modular polynomial with the smallest level for the given
2317 : * invariant, except if inv is INV_J, in which case return the modular
2318 : * polynomial of level L in {2,3,5}. NULL is returned if the modular
2319 : * polynomial can be calculated using polmodular0_powerup_ZM. */
2320 : INLINE GEN
2321 24848 : internal_db(long L, long inv)
2322 : {
2323 24848 : switch (inv) {
2324 23742 : case INV_J: switch (L) {
2325 19994 : case 2: return phi2_ZV();
2326 1889 : case 3: return phi3_ZV();
2327 1859 : case 5: return phi5_ZV();
2328 0 : default: break;
2329 : }
2330 182 : case INV_F: return phi5_f_ZV();
2331 14 : case INV_F2: return NULL;
2332 21 : case INV_F3: return phi3_f3_ZV();
2333 14 : case INV_F4: return NULL;
2334 119 : case INV_G2: return phi2_g2_ZV();
2335 56 : case INV_W2W3: return phi5_w2w3_ZV();
2336 14 : case INV_F8: return NULL;
2337 63 : case INV_W3W3: return phi5_w3w3_ZV();
2338 98 : case INV_W2W5: return phi7_w2w5_ZV();
2339 196 : case INV_W2W7: return phi3_w2w7_ZV();
2340 35 : case INV_W3W5: return phi2_w3w5_ZV();
2341 42 : case INV_W3W7: return phi5_w3w7_ZV();
2342 14 : case INV_W2W3E2: return NULL;
2343 21 : case INV_W2W5E2: return NULL;
2344 35 : case INV_W2W13: return phi3_w2w13_ZV();
2345 70 : case INV_W2W7E2: return NULL;
2346 21 : case INV_W3W3E2: return phi2_w3w3e2_ZV();
2347 56 : case INV_W5W7: return phi2_w5w7_ZV();
2348 14 : case INV_W3W13: return phi2_w3w13_ZV();
2349 21 : case INV_ATKIN3: return phi2_atkin3_ZV();
2350 : }
2351 0 : pari_err_BUG("internal_db");
2352 : return NULL;/*LCOV_EXCL_LINE*/
2353 : }
2354 :
2355 : /* NB: Should only be called if L <= modinv_max_internal_level(inv) */
2356 : static GEN
2357 24848 : polmodular_small_ZM(long L, long inv, GEN *db)
2358 : {
2359 24848 : GEN f = internal_db(L, inv);
2360 24848 : if (!f) return polmodular0_powerup_ZM(L, inv, db);
2361 24701 : return sympol_to_ZM(f, L);
2362 : }
2363 :
2364 : /* Each function phi_w?w?_j() returns a vector V containing two
2365 : * vectors u and v, and a scalar k, which together represent the
2366 : * bivariate polnomial
2367 : *
2368 : * phi(X, Y) = \sum_i u[i] X^i + Y \sum_i v[i] X^i + Y^2 X^k
2369 : */
2370 : static GEN
2371 1060 : phi_w2w3_j(void)
2372 : {
2373 : GEN phi, phi0, phi1;
2374 1060 : phi = cgetg(4, t_VEC);
2375 :
2376 1060 : phi0 = cgetg(14, t_VEC);
2377 1060 : gel(phi0, 1) = gen_1;
2378 1060 : gel(phi0, 2) = utoineg(0x3cUL);
2379 1060 : gel(phi0, 3) = utoi(0x702UL);
2380 1060 : gel(phi0, 4) = utoineg(0x797cUL);
2381 1060 : gel(phi0, 5) = utoi(0x5046fUL);
2382 1060 : gel(phi0, 6) = utoineg(0x1be0b8UL);
2383 1060 : gel(phi0, 7) = utoi(0x28ef9cUL);
2384 1060 : gel(phi0, 8) = utoi(0x15e2968UL);
2385 1060 : gel(phi0, 9) = utoi(0x1b8136fUL);
2386 1060 : gel(phi0, 10) = utoi(0xa67674UL);
2387 1060 : gel(phi0, 11) = utoi(0x23982UL);
2388 1060 : gel(phi0, 12) = utoi(0x294UL);
2389 1060 : gel(phi0, 13) = gen_1;
2390 :
2391 1060 : phi1 = cgetg(13, t_VEC);
2392 1060 : gel(phi1, 1) = gen_0;
2393 1060 : gel(phi1, 2) = gen_0;
2394 1060 : gel(phi1, 3) = gen_m1;
2395 1060 : gel(phi1, 4) = utoi(0x23UL);
2396 1060 : gel(phi1, 5) = utoineg(0xaeUL);
2397 1060 : gel(phi1, 6) = utoineg(0x5b8UL);
2398 1060 : gel(phi1, 7) = utoi(0x12d7UL);
2399 1060 : gel(phi1, 8) = utoineg(0x7c86UL);
2400 1060 : gel(phi1, 9) = utoi(0x37c8UL);
2401 1060 : gel(phi1, 10) = utoineg(0x69cUL);
2402 1060 : gel(phi1, 11) = utoi(0x48UL);
2403 1060 : gel(phi1, 12) = gen_m1;
2404 :
2405 1060 : gel(phi, 1) = phi0;
2406 1060 : gel(phi, 2) = phi1;
2407 1060 : gel(phi, 3) = utoi(5); return phi;
2408 : }
2409 :
2410 : static GEN
2411 4112 : phi_w3w3_j(void)
2412 : {
2413 : GEN phi, phi0, phi1;
2414 4112 : phi = cgetg(4, t_VEC);
2415 :
2416 4112 : phi0 = cgetg(14, t_VEC);
2417 4112 : gel(phi0, 1) = utoi(0x2d9UL);
2418 4112 : gel(phi0, 2) = utoi(0x4fbcUL);
2419 4112 : gel(phi0, 3) = utoi(0x5828aUL);
2420 4113 : gel(phi0, 4) = utoi(0x3a7a3cUL);
2421 4113 : gel(phi0, 5) = utoi(0x1bd8edfUL);
2422 4113 : gel(phi0, 6) = utoi(0x8348838UL);
2423 4113 : gel(phi0, 7) = utoi(0x1983f8acUL);
2424 4113 : gel(phi0, 8) = utoi(0x14e4e098UL);
2425 4113 : gel(phi0, 9) = utoi(0x69ed1a7UL);
2426 4113 : gel(phi0, 10) = utoi(0xc3828cUL);
2427 4113 : gel(phi0, 11) = utoi(0x2696aUL);
2428 4113 : gel(phi0, 12) = utoi(0x2acUL);
2429 4113 : gel(phi0, 13) = gen_1;
2430 :
2431 4113 : phi1 = cgetg(13, t_VEC);
2432 4113 : gel(phi1, 1) = gen_0;
2433 4113 : gel(phi1, 2) = utoineg(0x1bUL);
2434 4113 : gel(phi1, 3) = utoineg(0x5d6UL);
2435 4113 : gel(phi1, 4) = utoineg(0x1c7bUL);
2436 4113 : gel(phi1, 5) = utoi(0x7980UL);
2437 4113 : gel(phi1, 6) = utoi(0x12168UL);
2438 4113 : gel(phi1, 7) = utoineg(0x3528UL);
2439 4113 : gel(phi1, 8) = utoineg(0x6174UL);
2440 4113 : gel(phi1, 9) = utoi(0x2208UL);
2441 4113 : gel(phi1, 10) = utoineg(0x41dUL);
2442 4113 : gel(phi1, 11) = utoi(0x36UL);
2443 4113 : gel(phi1, 12) = gen_m1;
2444 :
2445 4113 : gel(phi, 1) = phi0;
2446 4113 : gel(phi, 2) = phi1;
2447 4113 : gel(phi, 3) = gen_2; return phi;
2448 : }
2449 :
2450 : static GEN
2451 3039 : phi_w2w5_j(void)
2452 : {
2453 : GEN phi, phi0, phi1;
2454 3039 : phi = cgetg(4, t_VEC);
2455 :
2456 3039 : phi0 = cgetg(20, t_VEC);
2457 3039 : gel(phi0, 1) = gen_1;
2458 3039 : gel(phi0, 2) = utoineg(0x2aUL);
2459 3039 : gel(phi0, 3) = utoi(0x549UL);
2460 3039 : gel(phi0, 4) = utoineg(0x6530UL);
2461 3039 : gel(phi0, 5) = utoi(0x60504UL);
2462 3039 : gel(phi0, 6) = utoineg(0x3cbbc8UL);
2463 3039 : gel(phi0, 7) = utoi(0x1d1ee74UL);
2464 3039 : gel(phi0, 8) = utoineg(0x7ef9ab0UL);
2465 3039 : gel(phi0, 9) = utoi(0x12b888beUL);
2466 3039 : gel(phi0, 10) = utoineg(0x15fa174cUL);
2467 3039 : gel(phi0, 11) = utoi(0x615d9feUL);
2468 3039 : gel(phi0, 12) = utoi(0xbeca070UL);
2469 3039 : gel(phi0, 13) = utoineg(0x88de74cUL);
2470 3039 : gel(phi0, 14) = utoineg(0x2b3a268UL);
2471 3039 : gel(phi0, 15) = utoi(0x24b3244UL);
2472 3039 : gel(phi0, 16) = utoi(0xb56270UL);
2473 3039 : gel(phi0, 17) = utoi(0x25989UL);
2474 3039 : gel(phi0, 18) = utoi(0x2a6UL);
2475 3039 : gel(phi0, 19) = gen_1;
2476 :
2477 3039 : phi1 = cgetg(19, t_VEC);
2478 3039 : gel(phi1, 1) = gen_0;
2479 3039 : gel(phi1, 2) = gen_0;
2480 3039 : gel(phi1, 3) = gen_m1;
2481 3039 : gel(phi1, 4) = utoi(0x1eUL);
2482 3039 : gel(phi1, 5) = utoineg(0xffUL);
2483 3039 : gel(phi1, 6) = utoi(0x243UL);
2484 3039 : gel(phi1, 7) = utoineg(0xf3UL);
2485 3039 : gel(phi1, 8) = utoineg(0x5c4UL);
2486 3039 : gel(phi1, 9) = utoi(0x107bUL);
2487 3039 : gel(phi1, 10) = utoineg(0x11b2fUL);
2488 3039 : gel(phi1, 11) = utoi(0x48fa8UL);
2489 3039 : gel(phi1, 12) = utoineg(0x6ff7cUL);
2490 3039 : gel(phi1, 13) = utoi(0x4bf48UL);
2491 3039 : gel(phi1, 14) = utoineg(0x187efUL);
2492 3039 : gel(phi1, 15) = utoi(0x404cUL);
2493 3039 : gel(phi1, 16) = utoineg(0x582UL);
2494 3039 : gel(phi1, 17) = utoi(0x3cUL);
2495 3039 : gel(phi1, 18) = gen_m1;
2496 :
2497 3039 : gel(phi, 1) = phi0;
2498 3039 : gel(phi, 2) = phi1;
2499 3039 : gel(phi, 3) = utoi(7); return phi;
2500 : }
2501 :
2502 : static GEN
2503 6651 : phi_w2w7_j(void)
2504 : {
2505 : GEN phi, phi0, phi1;
2506 6651 : phi = cgetg(4, t_VEC);
2507 :
2508 6651 : phi0 = cgetg(26, t_VEC);
2509 6650 : gel(phi0, 1) = gen_1;
2510 6650 : gel(phi0, 2) = utoineg(0x24UL);
2511 6650 : gel(phi0, 3) = utoi(0x4ceUL);
2512 6650 : gel(phi0, 4) = utoineg(0x5d60UL);
2513 6651 : gel(phi0, 5) = utoi(0x62b05UL);
2514 6651 : gel(phi0, 6) = utoineg(0x47be78UL);
2515 6651 : gel(phi0, 7) = utoi(0x2a3880aUL);
2516 6651 : gel(phi0, 8) = utoineg(0x114bccf4UL);
2517 6651 : gel(phi0, 9) = utoi(0x4b95e79aUL);
2518 6651 : gel(phi0, 10) = utoineg(0xe2cfee1cUL);
2519 6651 : gel(phi0, 11) = uu32toi(0x1UL, 0xe43d1126UL);
2520 6651 : gel(phi0, 12) = uu32toineg(0x2UL, 0xf04dc6f8UL);
2521 6651 : gel(phi0, 13) = uu32toi(0x3UL, 0x5384987dUL);
2522 6651 : gel(phi0, 14) = uu32toineg(0x2UL, 0xa5ccbe18UL);
2523 6651 : gel(phi0, 15) = uu32toi(0x1UL, 0x4c52c8a6UL);
2524 6651 : gel(phi0, 16) = utoineg(0x2643fdecUL);
2525 6651 : gel(phi0, 17) = utoineg(0x49f5ab66UL);
2526 6651 : gel(phi0, 18) = utoi(0x33074d3cUL);
2527 6651 : gel(phi0, 19) = utoineg(0x6a3e376UL);
2528 6651 : gel(phi0, 20) = utoineg(0x675aa58UL);
2529 6651 : gel(phi0, 21) = utoi(0x2674005UL);
2530 6651 : gel(phi0, 22) = utoi(0xba5be0UL);
2531 6651 : gel(phi0, 23) = utoi(0x2644eUL);
2532 6651 : gel(phi0, 24) = utoi(0x2acUL);
2533 6651 : gel(phi0, 25) = gen_1;
2534 :
2535 6651 : phi1 = cgetg(25, t_VEC);
2536 6651 : gel(phi1, 1) = gen_0;
2537 6651 : gel(phi1, 2) = gen_0;
2538 6651 : gel(phi1, 3) = gen_m1;
2539 6651 : gel(phi1, 4) = utoi(0x1cUL);
2540 6651 : gel(phi1, 5) = utoineg(0x10aUL);
2541 6651 : gel(phi1, 6) = utoi(0x3f0UL);
2542 6651 : gel(phi1, 7) = utoineg(0x5d3UL);
2543 6651 : gel(phi1, 8) = utoi(0x3efUL);
2544 6651 : gel(phi1, 9) = utoineg(0x102UL);
2545 6651 : gel(phi1, 10) = utoineg(0x5c8UL);
2546 6651 : gel(phi1, 11) = utoi(0x102fUL);
2547 6651 : gel(phi1, 12) = utoineg(0x13f8aUL);
2548 6651 : gel(phi1, 13) = utoi(0x86538UL);
2549 6651 : gel(phi1, 14) = utoineg(0x1bbd10UL);
2550 6651 : gel(phi1, 15) = utoi(0x3614e8UL);
2551 6651 : gel(phi1, 16) = utoineg(0x42f793UL);
2552 6651 : gel(phi1, 17) = utoi(0x364698UL);
2553 6651 : gel(phi1, 18) = utoineg(0x1c7a10UL);
2554 6651 : gel(phi1, 19) = utoi(0x97cc8UL);
2555 6651 : gel(phi1, 20) = utoineg(0x1fc8aUL);
2556 6651 : gel(phi1, 21) = utoi(0x4210UL);
2557 6651 : gel(phi1, 22) = utoineg(0x524UL);
2558 6651 : gel(phi1, 23) = utoi(0x38UL);
2559 6651 : gel(phi1, 24) = gen_m1;
2560 :
2561 6651 : gel(phi, 1) = phi0;
2562 6651 : gel(phi, 2) = phi1;
2563 6651 : gel(phi, 3) = utoi(9); return phi;
2564 : }
2565 :
2566 : static GEN
2567 2340 : phi_w2w13_j(void)
2568 : {
2569 : GEN phi, phi0, phi1;
2570 2340 : phi = cgetg(4, t_VEC);
2571 :
2572 2340 : phi0 = cgetg(44, t_VEC);
2573 2340 : gel(phi0, 1) = gen_1;
2574 2340 : gel(phi0, 2) = utoineg(0x1eUL);
2575 2340 : gel(phi0, 3) = utoi(0x45fUL);
2576 2340 : gel(phi0, 4) = utoineg(0x5590UL);
2577 2340 : gel(phi0, 5) = utoi(0x64407UL);
2578 2340 : gel(phi0, 6) = utoineg(0x53a792UL);
2579 2340 : gel(phi0, 7) = utoi(0x3b21af3UL);
2580 2340 : gel(phi0, 8) = utoineg(0x20d056d0UL);
2581 2340 : gel(phi0, 9) = utoi(0xe02db4a6UL);
2582 2340 : gel(phi0, 10) = uu32toineg(0x4UL, 0xb23400b0UL);
2583 2340 : gel(phi0, 11) = uu32toi(0x14UL, 0x57fbb906UL);
2584 2340 : gel(phi0, 12) = uu32toineg(0x49UL, 0xcf80c00UL);
2585 2340 : gel(phi0, 13) = uu32toi(0xdeUL, 0x84ff421UL);
2586 2340 : gel(phi0, 14) = uu32toineg(0x244UL, 0xc500c156UL);
2587 2340 : gel(phi0, 15) = uu32toi(0x52cUL, 0x79162979UL);
2588 2340 : gel(phi0, 16) = uu32toineg(0xa64UL, 0x8edc5650UL);
2589 2340 : gel(phi0, 17) = uu32toi(0x1289UL, 0x4225bb41UL);
2590 2340 : gel(phi0, 18) = uu32toineg(0x1d89UL, 0x2a15229aUL);
2591 2340 : gel(phi0, 19) = uu32toi(0x2a3eUL, 0x4539f1ebUL);
2592 2340 : gel(phi0, 20) = uu32toineg(0x366aUL, 0xa5ea1130UL);
2593 2340 : gel(phi0, 21) = uu32toi(0x3f47UL, 0xa19fecb4UL);
2594 2340 : gel(phi0, 22) = uu32toineg(0x4282UL, 0x91a3c4a0UL);
2595 2340 : gel(phi0, 23) = uu32toi(0x3f30UL, 0xbaa305b4UL);
2596 2340 : gel(phi0, 24) = uu32toineg(0x3635UL, 0xd11c2530UL);
2597 2340 : gel(phi0, 25) = uu32toi(0x29e2UL, 0x89df27ebUL);
2598 2340 : gel(phi0, 26) = uu32toineg(0x1d03UL, 0x6509d48aUL);
2599 2340 : gel(phi0, 27) = uu32toi(0x11e2UL, 0x272cc601UL);
2600 2340 : gel(phi0, 28) = uu32toineg(0x9b0UL, 0xacd58ff0UL);
2601 2340 : gel(phi0, 29) = uu32toi(0x485UL, 0x608d7db9UL);
2602 2340 : gel(phi0, 30) = uu32toineg(0x1bfUL, 0xa941546UL);
2603 2340 : gel(phi0, 31) = uu32toi(0x82UL, 0x56e48b21UL);
2604 2340 : gel(phi0, 32) = uu32toineg(0x13UL, 0xc36b2340UL);
2605 2340 : gel(phi0, 33) = uu32toineg(0x5UL, 0x6637257aUL);
2606 2340 : gel(phi0, 34) = uu32toi(0x5UL, 0x40f70bd0UL);
2607 2340 : gel(phi0, 35) = uu32toineg(0x1UL, 0xf70842daUL);
2608 2340 : gel(phi0, 36) = utoi(0x53eea5f0UL);
2609 2340 : gel(phi0, 37) = utoi(0xda17bf3UL);
2610 2340 : gel(phi0, 38) = utoineg(0xaf246c2UL);
2611 2340 : gel(phi0, 39) = utoi(0x278f847UL);
2612 2340 : gel(phi0, 40) = utoi(0xbf5550UL);
2613 2340 : gel(phi0, 41) = utoi(0x26f1fUL);
2614 2340 : gel(phi0, 42) = utoi(0x2b2UL);
2615 2340 : gel(phi0, 43) = gen_1;
2616 :
2617 2340 : phi1 = cgetg(43, t_VEC);
2618 2340 : gel(phi1, 1) = gen_0;
2619 2340 : gel(phi1, 2) = gen_0;
2620 2340 : gel(phi1, 3) = gen_m1;
2621 2340 : gel(phi1, 4) = utoi(0x1aUL);
2622 2340 : gel(phi1, 5) = utoineg(0x111UL);
2623 2340 : gel(phi1, 6) = utoi(0x5e4UL);
2624 2340 : gel(phi1, 7) = utoineg(0x1318UL);
2625 2340 : gel(phi1, 8) = utoi(0x2804UL);
2626 2340 : gel(phi1, 9) = utoineg(0x3cd6UL);
2627 2340 : gel(phi1, 10) = utoi(0x467cUL);
2628 2340 : gel(phi1, 11) = utoineg(0x3cd6UL);
2629 2340 : gel(phi1, 12) = utoi(0x2804UL);
2630 2340 : gel(phi1, 13) = utoineg(0x1318UL);
2631 2340 : gel(phi1, 14) = utoi(0x5e3UL);
2632 2340 : gel(phi1, 15) = utoineg(0x10dUL);
2633 2340 : gel(phi1, 16) = utoineg(0x5ccUL);
2634 2340 : gel(phi1, 17) = utoi(0x100bUL);
2635 2340 : gel(phi1, 18) = utoineg(0x160e1UL);
2636 2340 : gel(phi1, 19) = utoi(0xd2cb0UL);
2637 2340 : gel(phi1, 20) = utoineg(0x4c85fcUL);
2638 2340 : gel(phi1, 21) = utoi(0x137cb98UL);
2639 2340 : gel(phi1, 22) = utoineg(0x3c75568UL);
2640 2340 : gel(phi1, 23) = utoi(0x95c69c8UL);
2641 2340 : gel(phi1, 24) = utoineg(0x131557bcUL);
2642 2340 : gel(phi1, 25) = utoi(0x20aacfd0UL);
2643 2340 : gel(phi1, 26) = utoineg(0x2f9164e6UL);
2644 2340 : gel(phi1, 27) = utoi(0x3b6a5e40UL);
2645 2340 : gel(phi1, 28) = utoineg(0x3ff54344UL);
2646 2340 : gel(phi1, 29) = utoi(0x3b6a9140UL);
2647 2340 : gel(phi1, 30) = utoineg(0x2f927fa6UL);
2648 2340 : gel(phi1, 31) = utoi(0x20ae6450UL);
2649 2340 : gel(phi1, 32) = utoineg(0x131cd87cUL);
2650 2340 : gel(phi1, 33) = utoi(0x967d1e8UL);
2651 2340 : gel(phi1, 34) = utoineg(0x3d48ca8UL);
2652 2340 : gel(phi1, 35) = utoi(0x14333b8UL);
2653 2340 : gel(phi1, 36) = utoineg(0x5406bcUL);
2654 2340 : gel(phi1, 37) = utoi(0x10c130UL);
2655 2340 : gel(phi1, 38) = utoineg(0x27ba1UL);
2656 2340 : gel(phi1, 39) = utoi(0x433cUL);
2657 2340 : gel(phi1, 40) = utoineg(0x4c6UL);
2658 2340 : gel(phi1, 41) = utoi(0x34UL);
2659 2340 : gel(phi1, 42) = gen_m1;
2660 :
2661 2340 : gel(phi, 1) = phi0;
2662 2340 : gel(phi, 2) = phi1;
2663 2340 : gel(phi, 3) = utoi(15); return phi;
2664 : }
2665 :
2666 : static GEN
2667 1147 : phi_w3w5_j(void)
2668 : {
2669 : GEN phi, phi0, phi1;
2670 1147 : phi = cgetg(4, t_VEC);
2671 :
2672 1147 : phi0 = cgetg(26, t_VEC);
2673 1147 : gel(phi0, 1) = gen_1;
2674 1147 : gel(phi0, 2) = utoi(0x18UL);
2675 1147 : gel(phi0, 3) = utoi(0xb4UL);
2676 1147 : gel(phi0, 4) = utoineg(0x178UL);
2677 1147 : gel(phi0, 5) = utoineg(0x2d7eUL);
2678 1147 : gel(phi0, 6) = utoineg(0x89b8UL);
2679 1147 : gel(phi0, 7) = utoi(0x35c24UL);
2680 1147 : gel(phi0, 8) = utoi(0x128a18UL);
2681 1147 : gel(phi0, 9) = utoineg(0x12a911UL);
2682 1147 : gel(phi0, 10) = utoineg(0xcc0190UL);
2683 1147 : gel(phi0, 11) = utoi(0x94368UL);
2684 1147 : gel(phi0, 12) = utoi(0x1439d0UL);
2685 1147 : gel(phi0, 13) = utoi(0x96f931cUL);
2686 1147 : gel(phi0, 14) = utoineg(0x1f59ff0UL);
2687 1147 : gel(phi0, 15) = utoi(0x20e7e8UL);
2688 1147 : gel(phi0, 16) = utoineg(0x25fdf150UL);
2689 1147 : gel(phi0, 17) = utoineg(0x7091511UL);
2690 1147 : gel(phi0, 18) = utoi(0x1ef52f8UL);
2691 1147 : gel(phi0, 19) = utoi(0x341f2de4UL);
2692 1147 : gel(phi0, 20) = utoi(0x25d72c28UL);
2693 1147 : gel(phi0, 21) = utoi(0x95d2082UL);
2694 1147 : gel(phi0, 22) = utoi(0xd2d828UL);
2695 1147 : gel(phi0, 23) = utoi(0x281f4UL);
2696 1147 : gel(phi0, 24) = utoi(0x2b8UL);
2697 1147 : gel(phi0, 25) = gen_1;
2698 :
2699 1147 : phi1 = cgetg(25, t_VEC);
2700 1147 : gel(phi1, 1) = gen_0;
2701 1147 : gel(phi1, 2) = gen_0;
2702 1147 : gel(phi1, 3) = gen_0;
2703 1147 : gel(phi1, 4) = gen_1;
2704 1147 : gel(phi1, 5) = utoi(0xfUL);
2705 1147 : gel(phi1, 6) = utoi(0x2eUL);
2706 1147 : gel(phi1, 7) = utoineg(0x1fUL);
2707 1147 : gel(phi1, 8) = utoineg(0x2dUL);
2708 1147 : gel(phi1, 9) = utoineg(0x5caUL);
2709 1147 : gel(phi1, 10) = utoineg(0x358UL);
2710 1147 : gel(phi1, 11) = utoi(0x2f1cUL);
2711 1147 : gel(phi1, 12) = utoi(0xd8eaUL);
2712 1147 : gel(phi1, 13) = utoineg(0x38c70UL);
2713 1147 : gel(phi1, 14) = utoineg(0x1a964UL);
2714 1147 : gel(phi1, 15) = utoi(0x93512UL);
2715 1147 : gel(phi1, 16) = utoineg(0x58f2UL);
2716 1147 : gel(phi1, 17) = utoineg(0x5af1eUL);
2717 1147 : gel(phi1, 18) = utoi(0x1afb8UL);
2718 1147 : gel(phi1, 19) = utoi(0xc084UL);
2719 1147 : gel(phi1, 20) = utoineg(0x7fcbUL);
2720 1147 : gel(phi1, 21) = utoi(0x1c89UL);
2721 1147 : gel(phi1, 22) = utoineg(0x32aUL);
2722 1147 : gel(phi1, 23) = utoi(0x2dUL);
2723 1147 : gel(phi1, 24) = gen_m1;
2724 :
2725 1147 : gel(phi, 1) = phi0;
2726 1147 : gel(phi, 2) = phi1;
2727 1147 : gel(phi, 3) = utoi(8); return phi;
2728 : }
2729 :
2730 : static GEN
2731 2412 : phi_w3w7_j(void)
2732 : {
2733 : GEN phi, phi0, phi1;
2734 2412 : phi = cgetg(4, t_VEC);
2735 :
2736 2412 : phi0 = cgetg(34, t_VEC);
2737 2412 : gel(phi0, 1) = gen_1;
2738 2412 : gel(phi0, 2) = utoineg(0x14UL);
2739 2412 : gel(phi0, 3) = utoi(0x82UL);
2740 2412 : gel(phi0, 4) = utoi(0x1f8UL);
2741 2412 : gel(phi0, 5) = utoineg(0x2a45UL);
2742 2412 : gel(phi0, 6) = utoi(0x9300UL);
2743 2412 : gel(phi0, 7) = utoi(0x32abeUL);
2744 2412 : gel(phi0, 8) = utoineg(0x19c91cUL);
2745 2412 : gel(phi0, 9) = utoi(0xc1ba9UL);
2746 2412 : gel(phi0, 10) = utoi(0x1788f68UL);
2747 2412 : gel(phi0, 11) = utoineg(0x2b1989cUL);
2748 2412 : gel(phi0, 12) = utoineg(0x7a92408UL);
2749 2412 : gel(phi0, 13) = utoi(0x1238d56eUL);
2750 2412 : gel(phi0, 14) = utoi(0x13dd66a0UL);
2751 2412 : gel(phi0, 15) = utoineg(0x2dbedca8UL);
2752 2412 : gel(phi0, 16) = utoineg(0x34282eb8UL);
2753 2412 : gel(phi0, 17) = utoi(0x2c2a54d2UL);
2754 2412 : gel(phi0, 18) = utoi(0x98db81a8UL);
2755 2412 : gel(phi0, 19) = utoineg(0x4088be8UL);
2756 2412 : gel(phi0, 20) = utoineg(0xe424a220UL);
2757 2412 : gel(phi0, 21) = utoineg(0x67bbb232UL);
2758 2412 : gel(phi0, 22) = utoi(0x7dd8bb98UL);
2759 2412 : gel(phi0, 23) = uu32toi(0x1UL, 0xcaff744UL);
2760 2412 : gel(phi0, 24) = utoineg(0x1d46a378UL);
2761 2412 : gel(phi0, 25) = utoineg(0x82fa50f7UL);
2762 2412 : gel(phi0, 26) = utoineg(0x700ef38cUL);
2763 2412 : gel(phi0, 27) = utoi(0x20aa202eUL);
2764 2412 : gel(phi0, 28) = utoi(0x299b3440UL);
2765 2412 : gel(phi0, 29) = utoi(0xa476c4bUL);
2766 2412 : gel(phi0, 30) = utoi(0xd80558UL);
2767 2412 : gel(phi0, 31) = utoi(0x28a32UL);
2768 2412 : gel(phi0, 32) = utoi(0x2bcUL);
2769 2412 : gel(phi0, 33) = gen_1;
2770 :
2771 2412 : phi1 = cgetg(33, t_VEC);
2772 2412 : gel(phi1, 1) = gen_0;
2773 2412 : gel(phi1, 2) = gen_0;
2774 2412 : gel(phi1, 3) = gen_0;
2775 2412 : gel(phi1, 4) = gen_m1;
2776 2412 : gel(phi1, 5) = utoi(0xeUL);
2777 2412 : gel(phi1, 6) = utoineg(0x31UL);
2778 2412 : gel(phi1, 7) = utoineg(0xeUL);
2779 2412 : gel(phi1, 8) = utoi(0x99UL);
2780 2412 : gel(phi1, 9) = utoineg(0x8UL);
2781 2412 : gel(phi1, 10) = utoineg(0x2eUL);
2782 2412 : gel(phi1, 11) = utoineg(0x5ccUL);
2783 2412 : gel(phi1, 12) = utoi(0x308UL);
2784 2412 : gel(phi1, 13) = utoi(0x2904UL);
2785 2412 : gel(phi1, 14) = utoineg(0x15700UL);
2786 2412 : gel(phi1, 15) = utoineg(0x2b9ecUL);
2787 2412 : gel(phi1, 16) = utoi(0xf0966UL);
2788 2412 : gel(phi1, 17) = utoi(0xb3cc8UL);
2789 2412 : gel(phi1, 18) = utoineg(0x38241cUL);
2790 2412 : gel(phi1, 19) = utoineg(0x8604cUL);
2791 2412 : gel(phi1, 20) = utoi(0x578a64UL);
2792 2412 : gel(phi1, 21) = utoineg(0x11a798UL);
2793 2412 : gel(phi1, 22) = utoineg(0x39c85eUL);
2794 2412 : gel(phi1, 23) = utoi(0x1a5084UL);
2795 2412 : gel(phi1, 24) = utoi(0xcdeb4UL);
2796 2412 : gel(phi1, 25) = utoineg(0xb0364UL);
2797 2412 : gel(phi1, 26) = utoi(0x129d4UL);
2798 2412 : gel(phi1, 27) = utoi(0x126fcUL);
2799 2412 : gel(phi1, 28) = utoineg(0x8649UL);
2800 2412 : gel(phi1, 29) = utoi(0x1aa2UL);
2801 2412 : gel(phi1, 30) = utoineg(0x2dfUL);
2802 2412 : gel(phi1, 31) = utoi(0x2aUL);
2803 2412 : gel(phi1, 32) = gen_m1;
2804 :
2805 2412 : gel(phi, 1) = phi0;
2806 2412 : gel(phi, 2) = phi1;
2807 2412 : gel(phi, 3) = utoi(10); return phi;
2808 : }
2809 :
2810 : static GEN
2811 210 : phi_w3w13_j(void)
2812 : {
2813 : GEN phi, phi0, phi1;
2814 210 : phi = cgetg(4, t_VEC);
2815 :
2816 210 : phi0 = cgetg(58, t_VEC);
2817 210 : gel(phi0, 1) = gen_1;
2818 210 : gel(phi0, 2) = utoineg(0x10UL);
2819 210 : gel(phi0, 3) = utoi(0x58UL);
2820 210 : gel(phi0, 4) = utoi(0x258UL);
2821 210 : gel(phi0, 5) = utoineg(0x270cUL);
2822 210 : gel(phi0, 6) = utoi(0x9c00UL);
2823 210 : gel(phi0, 7) = utoi(0x2b40cUL);
2824 210 : gel(phi0, 8) = utoineg(0x20e250UL);
2825 210 : gel(phi0, 9) = utoi(0x4f46baUL);
2826 210 : gel(phi0, 10) = utoi(0x1869448UL);
2827 210 : gel(phi0, 11) = utoineg(0xa49ab68UL);
2828 210 : gel(phi0, 12) = utoi(0x96c7630UL);
2829 210 : gel(phi0, 13) = utoi(0x4f7e0af6UL);
2830 210 : gel(phi0, 14) = utoineg(0xea093590UL);
2831 210 : gel(phi0, 15) = utoineg(0x6735bc50UL);
2832 210 : gel(phi0, 16) = uu32toi(0x5UL, 0x971a2e08UL);
2833 210 : gel(phi0, 17) = uu32toineg(0x6UL, 0x29c9d965UL);
2834 210 : gel(phi0, 18) = uu32toineg(0xdUL, 0xeb9aa360UL);
2835 210 : gel(phi0, 19) = uu32toi(0x26UL, 0xe9c0584UL);
2836 210 : gel(phi0, 20) = uu32toineg(0x1UL, 0xb0cadce8UL);
2837 210 : gel(phi0, 21) = uu32toineg(0x62UL, 0x73586014UL);
2838 210 : gel(phi0, 22) = uu32toi(0x66UL, 0xaf672e38UL);
2839 210 : gel(phi0, 23) = uu32toi(0x6bUL, 0x93c28cdcUL);
2840 210 : gel(phi0, 24) = uu32toineg(0x11eUL, 0x4f633080UL);
2841 210 : gel(phi0, 25) = uu32toi(0x3cUL, 0xcc42461bUL);
2842 210 : gel(phi0, 26) = uu32toi(0x17bUL, 0xdec0a78UL);
2843 210 : gel(phi0, 27) = uu32toineg(0x166UL, 0x910d8bd0UL);
2844 210 : gel(phi0, 28) = uu32toineg(0xd4UL, 0x47873030UL);
2845 210 : gel(phi0, 29) = uu32toi(0x204UL, 0x811828baUL);
2846 210 : gel(phi0, 30) = uu32toineg(0x50UL, 0x5d713960UL);
2847 210 : gel(phi0, 31) = uu32toineg(0x198UL, 0xa27e42b0UL);
2848 210 : gel(phi0, 32) = uu32toi(0xe1UL, 0x25685138UL);
2849 210 : gel(phi0, 33) = uu32toi(0xe3UL, 0xaa5774bbUL);
2850 210 : gel(phi0, 34) = uu32toineg(0xcfUL, 0x392a9a00UL);
2851 210 : gel(phi0, 35) = uu32toineg(0x81UL, 0xfb334d04UL);
2852 210 : gel(phi0, 36) = uu32toi(0xabUL, 0x59594a68UL);
2853 210 : gel(phi0, 37) = uu32toi(0x42UL, 0x356993acUL);
2854 210 : gel(phi0, 38) = uu32toineg(0x86UL, 0x307ba678UL);
2855 210 : gel(phi0, 39) = uu32toineg(0xbUL, 0x7a9e59dcUL);
2856 210 : gel(phi0, 40) = uu32toi(0x4cUL, 0x27935f20UL);
2857 210 : gel(phi0, 41) = uu32toineg(0x2UL, 0xe0ac9045UL);
2858 210 : gel(phi0, 42) = uu32toineg(0x24UL, 0x14495758UL);
2859 210 : gel(phi0, 43) = utoi(0x20973410UL);
2860 210 : gel(phi0, 44) = uu32toi(0x13UL, 0x99ff4e00UL);
2861 210 : gel(phi0, 45) = uu32toineg(0x1UL, 0xa710d34aUL);
2862 210 : gel(phi0, 46) = uu32toineg(0x7UL, 0xfe5405c0UL);
2863 210 : gel(phi0, 47) = uu32toi(0x1UL, 0xcdee0f8UL);
2864 210 : gel(phi0, 48) = uu32toi(0x2UL, 0x660c92a8UL);
2865 210 : gel(phi0, 49) = utoi(0x3f13a35aUL);
2866 210 : gel(phi0, 50) = utoineg(0xe4eb4ba0UL);
2867 210 : gel(phi0, 51) = utoineg(0x6420f4UL);
2868 210 : gel(phi0, 52) = utoi(0x2c624370UL);
2869 210 : gel(phi0, 53) = utoi(0xb31b814UL);
2870 210 : gel(phi0, 54) = utoi(0xdd3ad8UL);
2871 210 : gel(phi0, 55) = utoi(0x29278UL);
2872 210 : gel(phi0, 56) = utoi(0x2c0UL);
2873 210 : gel(phi0, 57) = gen_1;
2874 :
2875 210 : phi1 = cgetg(57, t_VEC);
2876 210 : gel(phi1, 1) = gen_0;
2877 210 : gel(phi1, 2) = gen_0;
2878 210 : gel(phi1, 3) = gen_0;
2879 210 : gel(phi1, 4) = gen_m1;
2880 210 : gel(phi1, 5) = utoi(0xdUL);
2881 210 : gel(phi1, 6) = utoineg(0x34UL);
2882 210 : gel(phi1, 7) = utoi(0x1aUL);
2883 210 : gel(phi1, 8) = utoi(0xf7UL);
2884 210 : gel(phi1, 9) = utoineg(0x16cUL);
2885 210 : gel(phi1, 10) = utoineg(0xddUL);
2886 210 : gel(phi1, 11) = utoi(0x28aUL);
2887 210 : gel(phi1, 12) = utoineg(0xddUL);
2888 210 : gel(phi1, 13) = utoineg(0x16cUL);
2889 210 : gel(phi1, 14) = utoi(0xf6UL);
2890 210 : gel(phi1, 15) = utoi(0x1dUL);
2891 210 : gel(phi1, 16) = utoineg(0x31UL);
2892 210 : gel(phi1, 17) = utoineg(0x5ceUL);
2893 210 : gel(phi1, 18) = utoi(0x2e4UL);
2894 210 : gel(phi1, 19) = utoi(0x252cUL);
2895 210 : gel(phi1, 20) = utoineg(0x1b34cUL);
2896 210 : gel(phi1, 21) = utoi(0xaf80UL);
2897 210 : gel(phi1, 22) = utoi(0x1cc5f9UL);
2898 210 : gel(phi1, 23) = utoineg(0x3e1aa5UL);
2899 210 : gel(phi1, 24) = utoineg(0x86d17aUL);
2900 210 : gel(phi1, 25) = utoi(0x2427264UL);
2901 210 : gel(phi1, 26) = utoineg(0x691c1fUL);
2902 210 : gel(phi1, 27) = utoineg(0x862ad4eUL);
2903 210 : gel(phi1, 28) = utoi(0xab21e1fUL);
2904 210 : gel(phi1, 29) = utoi(0xbc19ddcUL);
2905 210 : gel(phi1, 30) = utoineg(0x24331db8UL);
2906 210 : gel(phi1, 31) = utoi(0x972c105UL);
2907 210 : gel(phi1, 32) = utoi(0x363d7107UL);
2908 210 : gel(phi1, 33) = utoineg(0x39696450UL);
2909 210 : gel(phi1, 34) = utoineg(0x1bce7c48UL);
2910 210 : gel(phi1, 35) = utoi(0x552ecba0UL);
2911 210 : gel(phi1, 36) = utoineg(0x1c7771b8UL);
2912 210 : gel(phi1, 37) = utoineg(0x393029b8UL);
2913 210 : gel(phi1, 38) = utoi(0x3755be97UL);
2914 210 : gel(phi1, 39) = utoi(0x83402a9UL);
2915 210 : gel(phi1, 40) = utoineg(0x24d5be62UL);
2916 210 : gel(phi1, 41) = utoi(0xdb6d90aUL);
2917 210 : gel(phi1, 42) = utoi(0xa0ef177UL);
2918 210 : gel(phi1, 43) = utoineg(0x99ff162UL);
2919 210 : gel(phi1, 44) = utoi(0xb09e27UL);
2920 210 : gel(phi1, 45) = utoi(0x26a7adcUL);
2921 210 : gel(phi1, 46) = utoineg(0x116e2fcUL);
2922 210 : gel(phi1, 47) = utoineg(0x1383b5UL);
2923 210 : gel(phi1, 48) = utoi(0x35a9e7UL);
2924 210 : gel(phi1, 49) = utoineg(0x1082a0UL);
2925 210 : gel(phi1, 50) = utoineg(0x4696UL);
2926 210 : gel(phi1, 51) = utoi(0x19f98UL);
2927 210 : gel(phi1, 52) = utoineg(0x8bb3UL);
2928 210 : gel(phi1, 53) = utoi(0x18bbUL);
2929 210 : gel(phi1, 54) = utoineg(0x297UL);
2930 210 : gel(phi1, 55) = utoi(0x27UL);
2931 210 : gel(phi1, 56) = gen_m1;
2932 :
2933 210 : gel(phi, 1) = phi0;
2934 210 : gel(phi, 2) = phi1;
2935 210 : gel(phi, 3) = utoi(16); return phi;
2936 : }
2937 :
2938 : static GEN
2939 2896 : phi_w5w7_j(void)
2940 : {
2941 : GEN phi, phi0, phi1;
2942 2896 : phi = cgetg(4, t_VEC);
2943 :
2944 2896 : phi0 = cgetg(50, t_VEC);
2945 2896 : gel(phi0, 1) = gen_1;
2946 2896 : gel(phi0, 2) = utoi(0xcUL);
2947 2896 : gel(phi0, 3) = utoi(0x2aUL);
2948 2896 : gel(phi0, 4) = utoi(0x10UL);
2949 2896 : gel(phi0, 5) = utoineg(0x69UL);
2950 2896 : gel(phi0, 6) = utoineg(0x318UL);
2951 2896 : gel(phi0, 7) = utoineg(0x148aUL);
2952 2896 : gel(phi0, 8) = utoineg(0x17c4UL);
2953 2896 : gel(phi0, 9) = utoi(0x1a73UL);
2954 2896 : gel(phi0, 10) = gen_0;
2955 2896 : gel(phi0, 11) = utoi(0x338a0UL);
2956 2896 : gel(phi0, 12) = utoi(0x61698UL);
2957 2896 : gel(phi0, 13) = utoineg(0x96e8UL);
2958 2896 : gel(phi0, 14) = utoi(0x140910UL);
2959 2896 : gel(phi0, 15) = utoineg(0x45f6b4UL);
2960 2896 : gel(phi0, 16) = utoineg(0x309f50UL);
2961 2896 : gel(phi0, 17) = utoineg(0xef9f8bUL);
2962 2896 : gel(phi0, 18) = utoineg(0x283167cUL);
2963 2896 : gel(phi0, 19) = utoi(0x625e20aUL);
2964 2896 : gel(phi0, 20) = utoineg(0x16186350UL);
2965 2896 : gel(phi0, 21) = utoi(0x46861281UL);
2966 2896 : gel(phi0, 22) = utoineg(0x754b96a0UL);
2967 2896 : gel(phi0, 23) = uu32toi(0x1UL, 0x421ca02aUL);
2968 2896 : gel(phi0, 24) = uu32toineg(0x2UL, 0xdb76a5cUL);
2969 2896 : gel(phi0, 25) = uu32toi(0x4UL, 0xf6afd8eUL);
2970 2896 : gel(phi0, 26) = uu32toineg(0x6UL, 0xaafd3cb4UL);
2971 2896 : gel(phi0, 27) = uu32toi(0x8UL, 0xda2539caUL);
2972 2896 : gel(phi0, 28) = uu32toineg(0xfUL, 0x84343790UL);
2973 2896 : gel(phi0, 29) = uu32toi(0xfUL, 0x914ff421UL);
2974 2896 : gel(phi0, 30) = uu32toineg(0x19UL, 0x3c123950UL);
2975 2896 : gel(phi0, 31) = uu32toi(0x15UL, 0x381f722aUL);
2976 2896 : gel(phi0, 32) = uu32toineg(0x15UL, 0xe01c0c24UL);
2977 2896 : gel(phi0, 33) = uu32toi(0x19UL, 0x3360b375UL);
2978 2896 : gel(phi0, 34) = utoineg(0x59fda9c0UL);
2979 2896 : gel(phi0, 35) = uu32toi(0x20UL, 0xff55024cUL);
2980 2896 : gel(phi0, 36) = uu32toi(0x16UL, 0xcc600800UL);
2981 2896 : gel(phi0, 37) = uu32toi(0x24UL, 0x1879c898UL);
2982 2896 : gel(phi0, 38) = uu32toi(0x1cUL, 0x37f97498UL);
2983 2896 : gel(phi0, 39) = uu32toi(0x19UL, 0x39ec4b60UL);
2984 2896 : gel(phi0, 40) = uu32toi(0x10UL, 0x52c660d0UL);
2985 2896 : gel(phi0, 41) = uu32toi(0x9UL, 0xcab00333UL);
2986 2896 : gel(phi0, 42) = uu32toi(0x4UL, 0x7fe69be4UL);
2987 2896 : gel(phi0, 43) = uu32toi(0x1UL, 0xa0c6f116UL);
2988 2896 : gel(phi0, 44) = utoi(0x69244638UL);
2989 2896 : gel(phi0, 45) = utoi(0xed560f7UL);
2990 2896 : gel(phi0, 46) = utoi(0xe7b660UL);
2991 2896 : gel(phi0, 47) = utoi(0x29d8aUL);
2992 2896 : gel(phi0, 48) = utoi(0x2c4UL);
2993 2896 : gel(phi0, 49) = gen_1;
2994 :
2995 2896 : phi1 = cgetg(49, t_VEC);
2996 2896 : gel(phi1, 1) = gen_0;
2997 2896 : gel(phi1, 2) = gen_0;
2998 2896 : gel(phi1, 3) = gen_0;
2999 2896 : gel(phi1, 4) = gen_0;
3000 2896 : gel(phi1, 5) = gen_0;
3001 2896 : gel(phi1, 6) = gen_1;
3002 2896 : gel(phi1, 7) = utoi(0x7UL);
3003 2896 : gel(phi1, 8) = utoi(0x8UL);
3004 2896 : gel(phi1, 9) = utoineg(0x9UL);
3005 2896 : gel(phi1, 10) = gen_0;
3006 2896 : gel(phi1, 11) = utoineg(0x13UL);
3007 2896 : gel(phi1, 12) = utoineg(0x7UL);
3008 2896 : gel(phi1, 13) = utoineg(0x5ceUL);
3009 2896 : gel(phi1, 14) = utoineg(0xb0UL);
3010 2896 : gel(phi1, 15) = utoi(0x460UL);
3011 2896 : gel(phi1, 16) = utoineg(0x194bUL);
3012 2896 : gel(phi1, 17) = utoi(0x87c3UL);
3013 2896 : gel(phi1, 18) = utoi(0x3cdeUL);
3014 2896 : gel(phi1, 19) = utoineg(0xd683UL);
3015 2896 : gel(phi1, 20) = utoi(0x6099bUL);
3016 2896 : gel(phi1, 21) = utoineg(0x111ea8UL);
3017 2896 : gel(phi1, 22) = utoi(0xfa113UL);
3018 2896 : gel(phi1, 23) = utoineg(0x1a6561UL);
3019 2896 : gel(phi1, 24) = utoineg(0x1e997UL);
3020 2896 : gel(phi1, 25) = utoi(0x214e54UL);
3021 2896 : gel(phi1, 26) = utoineg(0x29c3f4UL);
3022 2896 : gel(phi1, 27) = utoi(0x67e102UL);
3023 2896 : gel(phi1, 28) = utoineg(0x227eaaUL);
3024 2896 : gel(phi1, 29) = utoi(0x191d10UL);
3025 2896 : gel(phi1, 30) = utoi(0x1a9cd5UL);
3026 2896 : gel(phi1, 31) = utoineg(0x58386fUL);
3027 2896 : gel(phi1, 32) = utoi(0x2e49f6UL);
3028 2896 : gel(phi1, 33) = utoineg(0x31194bUL);
3029 2896 : gel(phi1, 34) = utoi(0x9e07aUL);
3030 2896 : gel(phi1, 35) = utoi(0x260d59UL);
3031 2896 : gel(phi1, 36) = utoineg(0x189921UL);
3032 2896 : gel(phi1, 37) = utoi(0xeca4aUL);
3033 2896 : gel(phi1, 38) = utoineg(0xa3d9cUL);
3034 2896 : gel(phi1, 39) = utoineg(0x426daUL);
3035 2896 : gel(phi1, 40) = utoi(0x91875UL);
3036 2896 : gel(phi1, 41) = utoineg(0x3b55bUL);
3037 2896 : gel(phi1, 42) = utoineg(0x56f4UL);
3038 2896 : gel(phi1, 43) = utoi(0xcd1bUL);
3039 2896 : gel(phi1, 44) = utoineg(0x5159UL);
3040 2896 : gel(phi1, 45) = utoi(0x10f4UL);
3041 2896 : gel(phi1, 46) = utoineg(0x20dUL);
3042 2896 : gel(phi1, 47) = utoi(0x23UL);
3043 2896 : gel(phi1, 48) = gen_m1;
3044 :
3045 2896 : gel(phi, 1) = phi0;
3046 2896 : gel(phi, 2) = phi1;
3047 2896 : gel(phi, 3) = utoi(12); return phi;
3048 : }
3049 :
3050 : static GEN
3051 3186 : phi_atkin3_j(void)
3052 : {
3053 : GEN phi, phi0, phi1;
3054 3186 : phi = cgetg(4, t_VEC);
3055 :
3056 3186 : phi0 = cgetg(6, t_VEC);
3057 3186 : gel(phi0, 1) = utoi(538141968);
3058 3186 : gel(phi0, 2) = utoi(19712160);
3059 3186 : gel(phi0, 3) = utoi(193752);
3060 3186 : gel(phi0, 4) = utoi(744);
3061 3186 : gel(phi0, 5) = gen_1;
3062 :
3063 3186 : phi1 = cgetg(5, t_VEC);
3064 3186 : gel(phi1, 1) = utoi(24528);
3065 3186 : gel(phi1, 2) = utoi(2348);
3066 3186 : gel(phi1, 3) = gen_0;
3067 3186 : gel(phi1, 4) = gen_m1;
3068 :
3069 3186 : gel(phi, 1) = phi0;
3070 3186 : gel(phi, 2) = phi1;
3071 3186 : gel(phi, 3) = gen_0; return phi;
3072 : }
3073 :
3074 : GEN
3075 27053 : double_eta_raw(long inv)
3076 : {
3077 27053 : switch (inv) {
3078 1060 : case INV_W2W3:
3079 1060 : case INV_W2W3E2: return phi_w2w3_j();
3080 4112 : case INV_W3W3:
3081 4112 : case INV_W3W3E2: return phi_w3w3_j();
3082 3039 : case INV_W2W5:
3083 3039 : case INV_W2W5E2: return phi_w2w5_j();
3084 6651 : case INV_W2W7:
3085 6651 : case INV_W2W7E2: return phi_w2w7_j();
3086 1147 : case INV_W3W5: return phi_w3w5_j();
3087 2412 : case INV_W3W7: return phi_w3w7_j();
3088 2340 : case INV_W2W13: return phi_w2w13_j();
3089 210 : case INV_W3W13: return phi_w3w13_j();
3090 2896 : case INV_W5W7: return phi_w5w7_j();
3091 3186 : case INV_ATKIN3: return phi_atkin3_j();
3092 : default: pari_err_BUG("double_eta_raw"); return NULL;/*LCOV_EXCL_LINE*/
3093 : }
3094 : }
3095 :
3096 : /* SECTION: Select discriminant for given modpoly level. */
3097 :
3098 : /* require an L1, useful for multi-threading */
3099 : #define MODPOLY_USE_L1 1
3100 : /* no bound on L1 other than the fixed bound MAX_L1 - needed to
3101 : * handle small L for certain invariants (but not for j) */
3102 : #define MODPOLY_NO_MAX_L1 2
3103 : /* don't use any auxilliary primes - needed to handle small L for
3104 : * certain invariants (but not for j) */
3105 : #define MODPOLY_NO_AUX_L 4
3106 : #define MODPOLY_IGNORE_SPARSE_FACTOR 8
3107 :
3108 : INLINE double
3109 3008 : modpoly_height_bound(long L, long inv)
3110 : {
3111 : double nbits, nbits2;
3112 : double c;
3113 : long hf;
3114 :
3115 : /* proven bound (in bits), derived from: 6l*log(l)+16*l+13*sqrt(l)*log(l) */
3116 3008 : nbits = 6.0*L*log2(L)+16/M_LN2*L+8.0*sqrt((double)L)*log2(L);
3117 : /* alternative proven bound (in bits), derived from: 6l*log(l)+17*l */
3118 3008 : nbits2 = 6.0*L*log2(L)+17/M_LN2*L;
3119 3008 : if ( nbits2 < nbits ) nbits = nbits2;
3120 3008 : hf = modinv_height_factor(inv);
3121 3008 : if (hf > 1) {
3122 : /* IMPORTANT: when dividing by the height factor, we only want to reduce
3123 : terms related to the bound on j (the roots of Phi_l(X,y)), not terms arising
3124 : from binomial coefficients. These arise in lemmas 2 and 3 of the height
3125 : bound paper, terms of (log 2)*L and 2.085*(L+1) which we convert here to
3126 : binary logs */
3127 : /* Massive overestimate: if you care about speed, determine a good height
3128 : * bound empirically as done for INV_F below */
3129 1641 : nbits2 = nbits - 4.01*L -3.0;
3130 1641 : nbits = nbits2/hf + 4.01*L + 3.0;
3131 : }
3132 3008 : if (inv == INV_F) {
3133 135 : if (L < 30) c = 45;
3134 35 : else if (L < 100) c = 36;
3135 21 : else if (L < 300) c = 32;
3136 7 : else if (L < 600) c = 26;
3137 0 : else if (L < 1200) c = 24;
3138 0 : else if (L < 2400) c = 22;
3139 0 : else c = 20;
3140 135 : nbits = (6.0*L*log2(L) + c*L)/hf;
3141 : }
3142 3008 : return nbits;
3143 : }
3144 :
3145 : /* small enough to write the factorization of a smooth in a BIL bit integer */
3146 : #define SMOOTH_PRIMES ((BITS_IN_LONG >> 1) - 1)
3147 :
3148 : #define MAX_ATKIN 255
3149 :
3150 : /* Must have primes at least up to MAX_ATKIN */
3151 : static const long PRIMES[] = {
3152 : 0, 2, 3, 5, 7, 11, 13, 17, 19, 23,
3153 : 29, 31, 37, 41, 43, 47, 53, 59, 61, 67,
3154 : 71, 73, 79, 83, 89, 97, 101, 103, 107, 109,
3155 : 113, 127, 131, 137, 139, 149, 151, 157, 163, 167,
3156 : 173, 179, 181, 191, 193, 197, 199, 211, 223, 227,
3157 : 229, 233, 239, 241, 251, 257, 263, 269, 271, 277
3158 : };
3159 :
3160 : #define MAX_L1 255
3161 :
3162 : typedef struct D_entry_struct {
3163 : ulong m;
3164 : long D, h;
3165 : } D_entry;
3166 :
3167 : /* Returns a form that generates the classes of norm p^2 in cl(p^2D)
3168 : * (i.e. one with order p-1), where p is an odd prime that splits in D
3169 : * and does not divide its conductor (but this is not verified) */
3170 : INLINE GEN
3171 77321 : qform_primeform2(long p, long D)
3172 : {
3173 77321 : GEN a = sqru(p), Dp2 = mulis(a, D), M = Z_factor(utoipos(p - 1));
3174 77321 : pari_sp av = avma;
3175 : long k;
3176 :
3177 156561 : for (k = D & 1; k <= p; k += 2)
3178 : {
3179 156561 : long ord, c = (k * k - D) / 4;
3180 : GEN Q, q;
3181 :
3182 156561 : if (!(c % p)) continue;
3183 134944 : q = mkqfis(a, k * p, c, Dp2); Q = qfbred_i(q);
3184 : /* TODO: How do we know that Q has order dividing p - 1? If we don't, then
3185 : * the call to gen_order should be replaced with a call to something with
3186 : * fastorder semantics (i.e. return 0 if ord(Q) \ndiv M). */
3187 134944 : ord = itos(qfi_order(Q, M));
3188 134944 : if (ord == p - 1) {
3189 : /* TODO: This check that gen_order returned the correct result should be
3190 : * removed when gen_order is replaced with fastorder semantics. */
3191 77321 : if (qfb_equal1(gpowgs(Q, p - 1))) return q;
3192 0 : break;
3193 : }
3194 57623 : set_avma(av);
3195 : }
3196 0 : return NULL;
3197 : }
3198 :
3199 : /* Let n = #cl(D); return x such that [L0]^x = [L] in cl(D), or -1 if x was
3200 : * not found */
3201 : INLINE long
3202 198033 : primeform_discrete_log(long L0, long L, long n, long D)
3203 : {
3204 198033 : pari_sp av = avma;
3205 198033 : GEN X, Q, R, DD = stoi(D);
3206 198033 : Q = primeform_u(DD, L0);
3207 198033 : R = primeform_u(DD, L);
3208 198033 : X = qfi_Shanks(R, Q, n);
3209 198033 : return gc_long(av, X? itos(X): -1);
3210 : }
3211 :
3212 : /* Return the norm of a class group generator appropriate for a discriminant
3213 : * that will be used to calculate the modular polynomial of level L and
3214 : * invariant inv. Don't consider norms less than initial_L0 */
3215 : static long
3216 3008 : select_L0(long L, long inv, long initial_L0)
3217 : {
3218 3008 : long L0, modinv_N = modinv_level(inv);
3219 :
3220 3008 : if (modinv_N % L == 0) pari_err_BUG("select_L0");
3221 :
3222 : /* TODO: Clean up these anomolous L0 choices */
3223 :
3224 : /* I've no idea why the discriminant-finding code fails with L0=5
3225 : * when L=19 and L=29, nor why L0=7 and L0=11 don't work for L=19
3226 : * either, nor why this happens for the otherwise unrelated
3227 : * invariants Weber-f and (2,3) double-eta. */
3228 3008 : if (inv == INV_W3W3E2 && L == 5) return 2;
3229 :
3230 2994 : if (inv == INV_F || inv == INV_F2 || inv == INV_F4 || inv == INV_F8
3231 2747 : || inv == INV_W2W3 || inv == INV_W2W3E2
3232 2684 : || inv == INV_W3W3 /* || inv == INV_W3W3E2 */) {
3233 429 : if (L == 19) return 13;
3234 386 : else if (L == 29 || L == 5) return 7;
3235 316 : return 5;
3236 : }
3237 2565 : if ((inv == INV_W2W5 || inv == INV_W2W5E2)
3238 126 : && (L == 7 || L == 19)) return 13;
3239 2530 : if ((inv == INV_W2W7 || inv == INV_W2W7E2)
3240 379 : && L == 11) return 13;
3241 2495 : if (inv == INV_W3W5) {
3242 63 : if (L == 7) return 13;
3243 56 : else if (L == 17) return 7;
3244 : }
3245 2488 : if (inv == INV_W3W7) {
3246 140 : if (L == 29 || L == 101) return 11;
3247 112 : if (L == 11 || L == 19) return 13;
3248 : }
3249 2432 : if (inv == INV_W5W7 && L == 17) return 3;
3250 :
3251 : /* L0 = smallest small prime different from L that doesn't divide modinv_N */
3252 2411 : for (L0 = unextprime(initial_L0 + 1);
3253 3357 : L0 == L || modinv_N % L0 == 0;
3254 946 : L0 = unextprime(L0 + 1))
3255 : ;
3256 2411 : return L0;
3257 : }
3258 :
3259 : /* Return the order of [L]^n in cl(D), where #cl(D) = ord. */
3260 : INLINE long
3261 1060162 : primeform_exp_order(long L, long n, long D, long ord)
3262 : {
3263 1060162 : pari_sp av = avma;
3264 1060162 : GEN Q = gpowgs(primeform_u(stoi(D), L), n);
3265 1060162 : long m = itos(qfi_order(Q, Z_factor(stoi(ord))));
3266 1060162 : return gc_long(av,m);
3267 : }
3268 :
3269 : /* If an ideal of norm modinv_deg is equivalent to an ideal of norm L0, we
3270 : * have an orientation ambiguity that we need to avoid. Note that we need to
3271 : * check all the possibilities (up to 8), but we can cheaply check inverses
3272 : * (so at most 2) */
3273 : static long
3274 34756 : orientation_ambiguity(long D1, long L0, long modinv_p1, long modinv_p2, long modinv_N)
3275 : {
3276 34756 : pari_sp av = avma;
3277 34756 : long ambiguity = 0;
3278 34756 : GEN D = stoi(D1), Q1 = primeform_u(D, modinv_p1), Q2 = NULL;
3279 :
3280 34756 : if (modinv_p2 > 1)
3281 : {
3282 32362 : if (modinv_p1 == modinv_p2) Q1 = gsqr(Q1);
3283 : else
3284 : {
3285 26525 : GEN P2 = primeform_u(D, modinv_p2);
3286 26525 : GEN Q = gsqr(P2), R = gsqr(Q1);
3287 : /* check that p1^2 != p2^{+/-2}, since this leads to
3288 : * ambiguities when converting j's to f's */
3289 26525 : if (equalii(gel(Q,1), gel(R,1)) && absequalii(gel(Q,2), gel(R,2)))
3290 : {
3291 0 : dbg_printf(3)("Bad D=%ld, a^2=b^2 problem between modinv_p1=%ld and modinv_p2=%ld\n",
3292 : D1, modinv_p1, modinv_p2);
3293 0 : ambiguity = 1;
3294 : }
3295 : else
3296 : { /* generate both p1*p2 and p1*p2^{-1} */
3297 26525 : Q2 = gmul(Q1, P2);
3298 26525 : P2 = ginv(P2);
3299 26525 : Q1 = gmul(Q1, P2);
3300 : }
3301 : }
3302 : }
3303 34756 : if (!ambiguity)
3304 : {
3305 34756 : GEN P = gsqr(primeform_u(D, L0));
3306 34756 : if (equalii(gel(P,1), gel(Q1,1))
3307 33726 : || (modinv_p2 > 1 && modinv_p1 != modinv_p2
3308 25509 : && equalii(gel(P,1), gel(Q2,1)))) {
3309 1480 : dbg_printf(3)("Bad D=%ld, a=b^{+/-2} problem between modinv_N=%ld and L0=%ld\n",
3310 : D1, modinv_N, L0);
3311 1480 : ambiguity = 1;
3312 : }
3313 : }
3314 34756 : return gc_long(av, ambiguity);
3315 : }
3316 :
3317 : static long
3318 770009 : check_generators(
3319 : long *n1_, long *m_,
3320 : long D, long h, long n, long subgrp_sz, long L0, long L1)
3321 : {
3322 770009 : long n1, m = primeform_exp_order(L0, n, D, h);
3323 770009 : if (m_) *m_ = m;
3324 770009 : n1 = n * m;
3325 770009 : if (!n1) pari_err_BUG("check_generators");
3326 770009 : *n1_ = n1;
3327 770009 : if (n1 < subgrp_sz/2 || ( ! L1 && n1 < subgrp_sz)) {
3328 28146 : dbg_printf(3)("Bad D1=%ld with n1=%ld, h1=%ld, L1=%ld: "
3329 : "L0 and L1 don't span subgroup of size d in cl(D1)\n",
3330 : D, n, h, L1);
3331 28146 : return 0;
3332 : }
3333 741863 : if (n1 < subgrp_sz && ! (n1 & 1)) {
3334 : int res;
3335 : /* check whether L1 is generated by L0, use the fact that it has order 2 */
3336 18159 : pari_sp av = avma;
3337 18159 : GEN D1 = stoi(D);
3338 18159 : GEN Q = gpowgs(primeform_u(D1, L0), n1 / 2);
3339 18159 : res = gequal(Q, qfbred_i(primeform_u(D1, L1)));
3340 18159 : set_avma(av);
3341 18159 : if (res) {
3342 5314 : dbg_printf(3)("Bad D1=%ld, with n1=%ld, h1=%ld, L1=%ld: "
3343 : "L1 generated by L0 in cl(D1)\n", D, n, h, L1);
3344 5314 : return 0;
3345 : }
3346 : }
3347 736549 : return 1;
3348 : }
3349 :
3350 : /* Calculate solutions (p, t) to the norm equation
3351 : * 4 p = t^2 - v^2 L^2 D (*)
3352 : * corresponding to the descriminant described by Dinfo.
3353 : *
3354 : * INPUT:
3355 : * - max: length of primes and traces
3356 : * - xprimes: p to exclude from primes (if they arise)
3357 : * - xcnt: length of xprimes
3358 : * - minbits: sum of log2(p) must be larger than this
3359 : * - Dinfo: discriminant, invariant and L for which we seek solutions to (*)
3360 : *
3361 : * OUTPUT:
3362 : * - primes: array of p in (*)
3363 : * - traces: array of t in (*)
3364 : * - totbits: sum of log2(p) for p in primes.
3365 : *
3366 : * RETURN:
3367 : * - the number of primes and traces found (these are always the same).
3368 : *
3369 : * NOTE: primes and traces are both NULL or both non-NULL.
3370 : * xprimes can be zero, in which case it is treated as empty. */
3371 : static long
3372 11928 : modpoly_pickD_primes(
3373 : ulong *primes, ulong *traces, long max, ulong *xprimes, long xcnt,
3374 : long *totbits, long minbits, disc_info *Dinfo)
3375 : {
3376 : double bits;
3377 : long D, m, n, vcnt, pfilter, one_prime, inv;
3378 : ulong maxp;
3379 : ulong a1, a2, v, t, p, a1_start, a1_delta, L0, L1, L, absD;
3380 11928 : ulong FF_BITS = BITS_IN_LONG - 2; /* BITS_IN_LONG - NAIL_BITS */
3381 :
3382 11928 : D = Dinfo->D1; absD = -D;
3383 11928 : L0 = Dinfo->L0;
3384 11928 : L1 = Dinfo->L1;
3385 11928 : L = Dinfo->L;
3386 11928 : inv = Dinfo->inv;
3387 :
3388 : /* make sure pfilter and D don't preclude the possibility of p=(t^2-v^2D)/4 being prime */
3389 11928 : pfilter = modinv_pfilter(inv);
3390 11928 : if ((pfilter & IQ_FILTER_1MOD3) && ! (D % 3)) return 0;
3391 11879 : if ((pfilter & IQ_FILTER_1MOD4) && ! (D & 0xF)) return 0;
3392 :
3393 : /* Naively estimate the number of primes satisfying 4p=t^2-L^2D with
3394 : * t=2 mod L and pfilter. This is roughly
3395 : * #{t: t^2 < max p and t=2 mod L} / pi(max p) * filter_density,
3396 : * where filter_density is 1, 2, or 4 depending on pfilter. If this quantity
3397 : * is already more than twice the number of bits we need, assume that,
3398 : * barring some obstruction, we should have no problem getting enough primes.
3399 : * In this case we just verify we can get one prime (which should always be
3400 : * true, assuming we chose D properly). */
3401 11879 : one_prime = 0;
3402 11879 : *totbits = 0;
3403 11879 : if (max <= 1 && ! one_prime) {
3404 8850 : p = ((pfilter & IQ_FILTER_1MOD3) ? 2 : 1) * ((pfilter & IQ_FILTER_1MOD4) ? 2 : 1);
3405 8850 : one_prime = (1UL << ((FF_BITS+1)/2)) * (log2(L*L*(-D))-1)
3406 8850 : > p*L*minbits*FF_BITS*M_LN2;
3407 8850 : if (one_prime) *totbits = minbits+1; /* lie */
3408 : }
3409 :
3410 11879 : m = n = 0;
3411 11879 : bits = 0.0;
3412 11879 : maxp = 0;
3413 29502 : for (v = 1; v < 100 && bits < minbits; v++) {
3414 : /* Don't allow v dividing the conductor. */
3415 26420 : if (ugcd(absD, v) != 1) continue;
3416 : /* Avoid v dividing the level. */
3417 26222 : if (v > 2 && modinv_is_double_eta(inv) && ugcd(modinv_level(inv), v) != 1)
3418 966 : continue;
3419 : /* can't get odd p with D=1 mod 8 unless v is even */
3420 25256 : if ((v & 1) && (D & 7) == 1) continue;
3421 : /* disallow 4 | v for L0=2 (removing this restriction is costly) */
3422 12509 : if (L0 == 2 && !(v & 3)) continue;
3423 : /* can't get p=3mod4 if v^2D is 0 mod 16 */
3424 12259 : if ((pfilter & IQ_FILTER_1MOD4) && !((v*v*D) & 0xF)) continue;
3425 12176 : if ((pfilter & IQ_FILTER_1MOD3) && !(v%3) ) continue;
3426 : /* avoid L0-volcanos with nonzero height */
3427 12118 : if (L0 != 2 && ! (v % L0)) continue;
3428 : /* ditto for L1 */
3429 12097 : if (L1 && !(v % L1)) continue;
3430 12097 : vcnt = 0;
3431 12097 : if ((v*v*absD)/4 > (1L<<FF_BITS)/(L*L)) break;
3432 12015 : if (both_odd(v,D)) {
3433 0 : a1_start = 1;
3434 0 : a1_delta = 2;
3435 : } else {
3436 12015 : a1_start = ((v*v*D) & 7)? 2: 0;
3437 12015 : a1_delta = 4;
3438 : }
3439 569011 : for (a1 = a1_start; bits < minbits; a1 += a1_delta) {
3440 565943 : a2 = (a1*a1 + v*v*absD) >> 2;
3441 565943 : if (!(a2 % L)) continue;
3442 478024 : t = a1*L + 2;
3443 478024 : p = a2*L*L + t - 1;
3444 : /* double check calculation just in case of overflow or other weirdness */
3445 478024 : if (!odd(p) || t*t + v*v*L*L*absD != 4*p)
3446 0 : pari_err_BUG("modpoly_pickD_primes");
3447 478024 : if (p > (1UL<<FF_BITS)) break;
3448 477792 : if (xprimes) {
3449 357303 : while (m < xcnt && xprimes[m] < p) m++;
3450 356875 : if (m < xcnt && p == xprimes[m]) {
3451 0 : dbg_printf(1)("skipping duplicate prime %ld\n", p);
3452 0 : continue;
3453 : }
3454 : }
3455 477792 : if (!modinv_good_prime(inv, p) || !uisprime(p)) continue;
3456 52969 : if (primes) {
3457 39057 : if (n >= max) goto done;
3458 : /* TODO: Implement test to filter primes that lead to
3459 : * L-valuation != 2 */
3460 39057 : primes[n] = p;
3461 39057 : traces[n] = t;
3462 : }
3463 52969 : n++;
3464 52969 : vcnt++;
3465 52969 : bits += log2(p);
3466 52969 : if (p > maxp) maxp = p;
3467 52969 : if (one_prime) goto done;
3468 : }
3469 3300 : if (vcnt)
3470 3297 : dbg_printf(3)("%ld primes with v=%ld, maxp=%ld (%.2f bits)\n",
3471 : vcnt, v, maxp, log2(maxp));
3472 : }
3473 3082 : done:
3474 11879 : if (!n) {
3475 9 : dbg_printf(3)("check_primes failed completely for D=%ld\n", D);
3476 9 : return 0;
3477 : }
3478 11870 : dbg_printf(3)("D=%ld: Found %ld primes totalling %0.2f of %ld bits\n",
3479 : D, n, bits, minbits);
3480 11870 : if (!*totbits) *totbits = (long)bits;
3481 11870 : return n;
3482 : }
3483 :
3484 : #define MAX_VOLCANO_FLOOR_SIZE 100000000
3485 :
3486 : static long
3487 3010 : calc_primes_for_discriminants(disc_info Ds[], long Dcnt, long L, long minbits)
3488 : {
3489 3010 : pari_sp av = avma;
3490 : long i, j, k, m, n, D1, pcnt, totbits;
3491 : ulong *primes, *Dprimes, *Dtraces;
3492 :
3493 : /* D1 is the discriminant with smallest absolute value among those we found */
3494 3010 : D1 = Ds[0].D1;
3495 8841 : for (i = 1; i < Dcnt; i++)
3496 5831 : if (Ds[i].D1 > D1) D1 = Ds[i].D1;
3497 :
3498 : /* n is an upper bound on the number of primes we might get. */
3499 3010 : n = ceil(minbits / (log2(L * L * (-D1)) - 2)) + 1;
3500 3010 : primes = (ulong *) stack_malloc(n * sizeof(*primes));
3501 3010 : Dprimes = (ulong *) stack_malloc(n * sizeof(*Dprimes));
3502 3010 : Dtraces = (ulong *) stack_malloc(n * sizeof(*Dtraces));
3503 3029 : for (i = 0, totbits = 0, pcnt = 0; i < Dcnt && totbits < minbits; i++)
3504 : {
3505 3029 : long np = modpoly_pickD_primes(Dprimes, Dtraces, n, primes, pcnt,
3506 3029 : &Ds[i].bits, minbits - totbits, Ds + i);
3507 3029 : ulong *T = (ulong *)newblock(2*np);
3508 3029 : Ds[i].nprimes = np;
3509 3029 : Ds[i].primes = T; memcpy(T , Dprimes, np * sizeof(*Dprimes));
3510 3029 : Ds[i].traces = T+np; memcpy(T+np, Dtraces, np * sizeof(*Dtraces));
3511 :
3512 3029 : totbits += Ds[i].bits;
3513 3029 : pcnt += np;
3514 :
3515 3029 : if (totbits >= minbits || i == Dcnt - 1) { Dcnt = i + 1; break; }
3516 : /* merge lists */
3517 599 : for (j = pcnt - np - 1, k = np - 1, m = pcnt - 1; m >= 0; m--) {
3518 580 : if (k >= 0) {
3519 555 : if (j >= 0 && primes[j] > Dprimes[k])
3520 301 : primes[m] = primes[j--];
3521 : else
3522 254 : primes[m] = Dprimes[k--];
3523 : } else {
3524 25 : primes[m] = primes[j--];
3525 : }
3526 : }
3527 : }
3528 3010 : if (totbits < minbits) {
3529 2 : dbg_printf(1)("Only obtained %ld of %ld bits using %ld discriminants\n",
3530 : totbits, minbits, Dcnt);
3531 4 : for (i = 0; i < Dcnt; i++) killblock((GEN)Ds[i].primes);
3532 2 : Dcnt = 0;
3533 : }
3534 3010 : return gc_long(av, Dcnt);
3535 : }
3536 :
3537 : /* Select discriminant(s) to use when calculating the modular
3538 : * polynomial of level L and invariant inv.
3539 : *
3540 : * INPUT:
3541 : * - L: level of modular polynomial (must be odd)
3542 : * - inv: invariant of modular polynomial
3543 : * - L0: result of select_L0(L, inv)
3544 : * - minbits: height of modular polynomial
3545 : * - flags: see below
3546 : * - tab: result of scanD0(L0)
3547 : * - tablen: length of tab
3548 : *
3549 : * OUTPUT:
3550 : * - Ds: the selected discriminant(s)
3551 : *
3552 : * RETURN:
3553 : * - the number of Ds found
3554 : *
3555 : * The flags parameter is constructed by ORing zero or more of the
3556 : * following values:
3557 : * - MODPOLY_USE_L1: force use of second class group generator
3558 : * - MODPOLY_NO_AUX_L: don't use auxillary class group elements
3559 : * - MODPOLY_IGNORE_SPARSE_FACTOR: obtain D for which h(D) > L + 1
3560 : * rather than h(D) > (L + 1)/s */
3561 : static long
3562 3010 : modpoly_pickD(disc_info Ds[MODPOLY_MAX_DCNT], long L, long inv,
3563 : long L0, long max_L1, long minbits, long flags, D_entry *tab, long tablen)
3564 : {
3565 3010 : pari_sp ltop = avma, btop;
3566 : disc_info Dinfo;
3567 : pari_timer T;
3568 : long modinv_p1, modinv_p2; /* const after next line */
3569 3010 : const long modinv_deg = modinv_degree(&modinv_p1, &modinv_p2, inv);
3570 3010 : const long pfilter = modinv_pfilter(inv), modinv_N = modinv_level(inv);
3571 : long i, k, use_L1, Dcnt, D0_i, d, cost, enum_cost, best_cost, totbits;
3572 3010 : const double L_bits = log2(L);
3573 :
3574 3010 : if (!odd(L)) pari_err_BUG("modpoly_pickD");
3575 :
3576 3010 : timer_start(&T);
3577 3010 : if (flags & MODPOLY_IGNORE_SPARSE_FACTOR) d = L+2;
3578 2863 : else d = ceildivuu(L+1, modinv_sparse_factor(inv)) + 1;
3579 :
3580 : /* Now set level to 0 unless we will need to compute N-isogenies */
3581 3010 : dbg_printf(1)("Using L0=%ld for L=%ld, d=%ld, modinv_N=%ld, modinv_deg=%ld\n",
3582 : L0, L, d, modinv_N, modinv_deg);
3583 :
3584 : /* We use L1 if (L0|L) == 1 or if we are forced to by flags. */
3585 3010 : use_L1 = (kross(L0,L) > 0 || (flags & MODPOLY_USE_L1));
3586 :
3587 3010 : Dcnt = best_cost = totbits = 0;
3588 3010 : dbg_printf(3)("use_L1=%ld\n", use_L1);
3589 3010 : dbg_printf(3)("minbits = %ld\n", minbits);
3590 :
3591 : /* Iterate over the fundamental discriminants for L0 */
3592 1852574 : for (D0_i = 0; D0_i < tablen; D0_i++)
3593 : {
3594 1849564 : D_entry D0_entry = tab[D0_i];
3595 1849564 : long m, n0, h0, deg, L1, H_cost, twofactor, D0 = D0_entry.D;
3596 : double D0_bits;
3597 2878918 : if (! modinv_good_disc(inv, D0)) continue;
3598 1238991 : dbg_printf(3)("D0=%ld\n", D0);
3599 : /* don't allow either modinv_p1 or modinv_p2 to ramify */
3600 1238991 : if (kross(D0, L) < 1
3601 559663 : || (modinv_p1 > 1 && kross(D0, modinv_p1) < 1)
3602 549952 : || (modinv_p2 > 1 && kross(D0, modinv_p2) < 1)) {
3603 699212 : dbg_printf(3)("Bad D0=%ld due to nonsplit L or ramified level\n", D0);
3604 699212 : continue;
3605 : }
3606 539779 : deg = D0_entry.h; /* class poly degree */
3607 539779 : h0 = ((D0_entry.m & 2) ? 2*deg : deg); /* class number */
3608 : /* (D0_entry.m & 1) is 1 if ord(L0) < h0 (hence = h0/2),
3609 : * is 0 if ord(L0) = h0 */
3610 539779 : n0 = h0 / ((D0_entry.m & 1) + 1); /* = ord(L0) */
3611 :
3612 : /* Look for L1: for each smooth prime p */
3613 539779 : L1 = 0;
3614 13112844 : for (i = 1 ; i <= SMOOTH_PRIMES; i++)
3615 : {
3616 12683544 : long p = PRIMES[i];
3617 12683544 : if (p <= L0) continue;
3618 : /* If 1 + (D0 | p) = 1, i.e. p | D0 */
3619 11964772 : if (((D0_entry.m >> (2*i)) & 3) == 1) {
3620 : /* XXX: Why (p | L) = -1? Presumably so (L^2 v^2 D0 | p) = -1? */
3621 390736 : if (p <= max_L1 && modinv_N % p && kross(p,L) < 0) { L1 = p; break; }
3622 : }
3623 : }
3624 539779 : if (i > SMOOTH_PRIMES && (n0 < h0 || use_L1))
3625 : { /* Didn't find suitable L1 though we need one */
3626 249626 : dbg_printf(3)("Bad D0=%ld because there is no good L1\n", D0);
3627 249626 : continue;
3628 : }
3629 290153 : dbg_printf(3)("Good D0=%ld with L1=%ld, n0=%ld, h0=%ld, d=%ld\n",
3630 : D0, L1, n0, h0, d);
3631 :
3632 : /* We're finished if we have sufficiently many discriminants that satisfy
3633 : * the cost requirement */
3634 290153 : if (totbits > minbits && best_cost && h0*(L-1) > 3*best_cost) break;
3635 :
3636 290153 : D0_bits = log2(-D0);
3637 : /* If L^2 D0 is too big to fit in a BIL bit integer, skip D0. */
3638 290153 : if (D0_bits + 2 * L_bits > (BITS_IN_LONG - 1)) continue;
3639 :
3640 : /* m is the order of L0^n0 in L^2 D0? */
3641 290153 : m = primeform_exp_order(L0, n0, L * L * D0, n0 * (L-1));
3642 290153 : if (m < (L-1)/2) {
3643 80516 : dbg_printf(3)("Bad D0=%ld because %ld is less than (L-1)/2=%ld\n",
3644 0 : D0, m, (L - 1)/2);
3645 80516 : continue;
3646 : }
3647 : /* Heuristic. Doesn't end up contributing much. */
3648 209637 : H_cost = 2 * deg * deg;
3649 :
3650 : /* 0xc = 0b1100, so D0_entry.m & 0xc == 1 + (D0 | 2) */
3651 209637 : if ((D0 & 7) == 5) /* D0 = 5 (mod 8) */
3652 5897 : twofactor = ((D0_entry.m & 0xc) ? 1 : 3);
3653 : else
3654 203740 : twofactor = 0;
3655 :
3656 209637 : btop = avma;
3657 : /* For each small prime... */
3658 722968 : for (i = 0; i <= SMOOTH_PRIMES; i++) {
3659 : long h1, h2, D1, D2, n1, n2, dl1, dl20, dl21, p, q, j;
3660 : double p_bits;
3661 722863 : set_avma(btop);
3662 : /* i = 0 corresponds to 1, which we do not want to skip! (i.e. DK = D) */
3663 722863 : if (i) {
3664 1016717 : if (modinv_odd_conductor(inv) && i == 1) continue;
3665 503933 : p = PRIMES[i];
3666 : /* Don't allow large factors in the conductor. */
3667 619145 : if (p > max_L1) break;
3668 409613 : if (p == L0 || p == L1 || p == L || p == modinv_p1 || p == modinv_p2)
3669 141194 : continue;
3670 268419 : p_bits = log2(p);
3671 : /* h1 is the class number of D1 = q^2 D0, where q = p^j (j defined in the loop below) */
3672 268419 : h1 = h0 * (p - ((D0_entry.m >> (2*i)) & 0x3) + 1);
3673 : /* q is the smallest power of p such that h1 >= d ~ "L + 1". */
3674 271143 : for (j = 1, q = p; h1 < d; j++, q *= p, h1 *= p)
3675 : ;
3676 268419 : D1 = q * q * D0;
3677 : /* can't have D1 = 0 mod 16 and hope to get any primes congruent to 3 mod 4 */
3678 268419 : if ((pfilter & IQ_FILTER_1MOD4) && !(D1 & 0xF)) continue;
3679 : } else {
3680 : /* i = 0, corresponds to "p = 1". */
3681 209637 : h1 = h0;
3682 209637 : D1 = D0;
3683 209637 : p = q = j = 1;
3684 209637 : p_bits = 0;
3685 : }
3686 : /* include a factor of 4 if D1 is 5 mod 8 */
3687 : /* XXX: No idea why he does this. */
3688 477986 : if (twofactor && (q & 1)) {
3689 14046 : if (modinv_odd_conductor(inv)) continue;
3690 518 : D1 *= 4;
3691 518 : h1 *= twofactor;
3692 : }
3693 : /* heuristic early abort; we may miss good D1's, but this saves time */
3694 464458 : if (totbits > minbits && best_cost && h1*(L-1) > 2.2*best_cost) continue;
3695 :
3696 : /* log2(D0 * (p^j)^2 * L^2 * twofactor) > (BIL - 1) -- params too big. */
3697 911300 : if (D0_bits + 2*j*p_bits + 2*L_bits
3698 454804 : + (twofactor && (q & 1) ? 2.0 : 0.0) > (BITS_IN_LONG-1)) continue;
3699 :
3700 453112 : if (! check_generators(&n1, NULL, D1, h1, n0, d, L0, L1)) continue;
3701 :
3702 435213 : if (n1 >= h1) dl1 = -1; /* fill it in later */
3703 195051 : else if ((dl1 = primeform_discrete_log(L0, L, n1, D1)) < 0) continue;
3704 318377 : dbg_printf(3)("Good D0=%ld, D1=%ld with q=%ld, L1=%ld, n1=%ld, h1=%ld\n",
3705 : D0, D1, q, L1, n1, h1);
3706 318377 : if (modinv_deg && orientation_ambiguity(D1, L0, modinv_p1, modinv_p2, modinv_N))
3707 1480 : continue;
3708 :
3709 316897 : D2 = L * L * D1;
3710 316897 : h2 = h1 * (L-1);
3711 : /* m is the order of L0^n1 in cl(D2) */
3712 316897 : if (!check_generators(&n2, &m, D2, h2, n1, d*(L-1), L0, L1)) continue;
3713 :
3714 : /* This restriction on m is not necessary, but simplifies life later */
3715 301336 : if (m < (L-1)/2 || (!L1 && m < L-1)) {
3716 147733 : dbg_printf(3)("Bad D2=%ld for D1=%ld, D0=%ld, with n2=%ld, h2=%ld, L1=%ld, "
3717 : "order of L0^n1 in cl(D2) is too small\n", D2, D1, D0, n2, h2, L1);
3718 147733 : continue;
3719 : }
3720 153603 : dl20 = n1;
3721 153603 : dl21 = 0;
3722 153603 : if (m < L-1) {
3723 77321 : GEN Q1 = qform_primeform2(L, D1), Q2, X;
3724 77321 : if (!Q1) pari_err_BUG("modpoly_pickD");
3725 77321 : Q2 = primeform_u(stoi(D2), L1);
3726 77321 : Q2 = qfbcomp(Q1, Q2); /* we know this element has order L-1 */
3727 77321 : Q1 = primeform_u(stoi(D2), L0);
3728 77321 : k = ((n2 & 1) ? 2*n2 : n2)/(L-1);
3729 77321 : Q1 = gpowgs(Q1, k);
3730 77321 : X = qfi_Shanks(Q2, Q1, L-1);
3731 77321 : if (!X) {
3732 11607 : dbg_printf(3)("Bad D2=%ld for D1=%ld, D0=%ld, with n2=%ld, h2=%ld, L1=%ld, "
3733 : "form of norm L^2 not generated by L0 and L1\n",
3734 : D2, D1, D0, n2, h2, L1);
3735 11607 : continue;
3736 : }
3737 65714 : dl20 = itos(X) * k;
3738 65714 : dl21 = 1;
3739 : }
3740 141996 : if (! (m < L-1 || n2 < d*(L-1)) && n1 >= d && ! use_L1)
3741 75748 : L1 = 0; /* we don't need L1 */
3742 :
3743 141996 : if (!L1 && use_L1) {
3744 0 : dbg_printf(3)("not using D2=%ld for D1=%ld, D0=%ld, with n2=%ld, h2=%ld, L1=%ld, "
3745 : "because we don't need L1 but must use it\n",
3746 : D2, D1, D0, n2, h2, L1);
3747 0 : continue;
3748 : }
3749 : /* don't allow zero dl21 with L1 for the moment, since
3750 : * modpoly doesn't handle it - we may change this in the future */
3751 141996 : if (L1 && ! dl21) continue;
3752 141462 : dbg_printf(3)("Good D0=%ld, D1=%ld, D2=%ld with s=%ld^%ld, L1=%ld, dl2=%ld, n2=%ld, h2=%ld\n",
3753 : D0, D1, D2, p, j, L1, dl20, n2, h2);
3754 :
3755 : /* This estimate is heuristic and fiddling with the
3756 : * parameters 5 and 0.25 can change things quite a bit. */
3757 141462 : enum_cost = n2 * (5 * L0 * L0 + 0.25 * L1 * L1);
3758 141462 : cost = enum_cost + H_cost;
3759 141462 : if (best_cost && cost > 2.2*best_cost) break;
3760 34544 : if (best_cost && cost >= 0.99*best_cost) continue;
3761 :
3762 8899 : Dinfo.GENcode0 = evaltyp(t_VECSMALL)|_evallg(13);
3763 8899 : Dinfo.inv = inv;
3764 8899 : Dinfo.L = L;
3765 8899 : Dinfo.D0 = D0;
3766 8899 : Dinfo.D1 = D1;
3767 8899 : Dinfo.L0 = L0;
3768 8899 : Dinfo.L1 = L1;
3769 8899 : Dinfo.n1 = n1;
3770 8899 : Dinfo.n2 = n2;
3771 8899 : Dinfo.dl1 = dl1;
3772 8899 : Dinfo.dl2_0 = dl20;
3773 8899 : Dinfo.dl2_1 = dl21;
3774 8899 : Dinfo.cost = cost;
3775 :
3776 8899 : if (!modpoly_pickD_primes(NULL, NULL, 0, NULL, 0, &Dinfo.bits, minbits, &Dinfo))
3777 58 : continue;
3778 8841 : dbg_printf(2)("Best D2=%ld, D1=%ld, D0=%ld with s=%ld^%ld, L1=%ld, "
3779 : "n1=%ld, n2=%ld, cost ratio %.2f, bits=%ld\n",
3780 : D2, D1, D0, p, j, L1, n1, n2,
3781 0 : (double)cost/(d*(L-1)), Dinfo.bits);
3782 : /* Insert Dinfo into the Ds array. Ds is sorted by ascending cost. */
3783 47897 : for (j = 0; j < Dcnt; j++)
3784 44876 : if (Dinfo.cost < Ds[j].cost) break;
3785 8841 : if (n2 > MAX_VOLCANO_FLOOR_SIZE && n2*(L1 ? 2 : 1) > 1.2* (d*(L-1)) ) {
3786 0 : dbg_printf(3)("Not using D1=%ld, D2=%ld for space reasons\n", D1, D2);
3787 0 : continue;
3788 : }
3789 8841 : if (j == Dcnt && Dcnt == MODPOLY_MAX_DCNT)
3790 0 : continue;
3791 8841 : totbits += Dinfo.bits;
3792 8841 : if (Dcnt == MODPOLY_MAX_DCNT) totbits -= Ds[Dcnt-1].bits;
3793 8841 : if (Dcnt < MODPOLY_MAX_DCNT) Dcnt++;
3794 8841 : if (n2 > MAX_VOLCANO_FLOOR_SIZE)
3795 0 : dbg_printf(3)("totbits=%ld, minbits=%ld\n", totbits, minbits);
3796 19674 : for (k = Dcnt-1; k > j; k--) Ds[k] = Ds[k-1];
3797 8841 : Ds[k] = Dinfo;
3798 8841 : best_cost = (totbits > minbits)? Ds[Dcnt-1].cost: 0;
3799 : /* if we were able to use D1 with s = 1, there is no point in
3800 : * using any larger D1 for the same D0 */
3801 8841 : if (!i) break;
3802 : } /* END FOR over small primes */
3803 : } /* END WHILE over D0's */
3804 3010 : dbg_printf(2)(" checked %ld of %ld fundamental discriminants to find suitable "
3805 : "discriminant (Dcnt = %ld)\n", D0_i, tablen, Dcnt);
3806 3010 : if ( ! Dcnt) {
3807 0 : dbg_printf(1)("failed completely for L=%ld\n", L);
3808 0 : return 0;
3809 : }
3810 :
3811 3010 : Dcnt = calc_primes_for_discriminants(Ds, Dcnt, L, minbits);
3812 :
3813 : /* fill in any missing dl1's */
3814 6037 : for (i = 0 ; i < Dcnt; i++)
3815 3027 : if (Ds[i].dl1 < 0 &&
3816 2982 : (Ds[i].dl1 = primeform_discrete_log(L0, L, Ds[i].n1, Ds[i].D1)) < 0)
3817 0 : pari_err_BUG("modpoly_pickD");
3818 3010 : if (DEBUGLEVEL > 1+3) {
3819 0 : err_printf("Selected %ld discriminants using %ld msecs\n", Dcnt, timer_delay(&T));
3820 0 : for (i = 0 ; i < Dcnt ; i++)
3821 : {
3822 0 : GEN H = classno(stoi(Ds[i].D0));
3823 0 : long h0 = itos(H);
3824 0 : err_printf (" D0=%ld, h(D0)=%ld, D=%ld, L0=%ld, L1=%ld, "
3825 : "cost ratio=%.2f, enum ratio=%.2f,",
3826 0 : Ds[i].D0, h0, Ds[i].D1, Ds[i].L0, Ds[i].L1,
3827 0 : (double)Ds[i].cost/(d*(L-1)),
3828 0 : (double)(Ds[i].n2*(Ds[i].L1 ? 2 : 1))/(d*(L-1)));
3829 0 : err_printf (" %ld primes, %ld bits\n", Ds[i].nprimes, Ds[i].bits);
3830 : }
3831 : }
3832 3010 : return gc_long(ltop, Dcnt);
3833 : }
3834 :
3835 : static int
3836 14433820 : _qsort_cmp(const void *a, const void *b)
3837 : {
3838 14433820 : D_entry *x = (D_entry *)a, *y = (D_entry *)b;
3839 : long u, v;
3840 :
3841 : /* u and v are the class numbers of x and y */
3842 14433820 : u = x->h * (!!(x->m & 2) + 1);
3843 14433820 : v = y->h * (!!(y->m & 2) + 1);
3844 : /* Sort by class number */
3845 14433820 : if (u < v) return -1;
3846 10045490 : if (u > v) return 1;
3847 : /* Sort by discriminant (which is < 0, hence the sign reversal) */
3848 3015732 : if (x->D > y->D) return -1;
3849 0 : if (x->D < y->D) return 1;
3850 0 : return 0;
3851 : }
3852 :
3853 : /* Build a table containing fundamental D, |D| <= maxD whose class groups
3854 : * - are cyclic generated by an element of norm L0
3855 : * - have class number at most maxh
3856 : * The table is ordered using _qsort_cmp above, which ranks the discriminants
3857 : * by class number, then by absolute discriminant.
3858 : *
3859 : * INPUT:
3860 : * - maxd: largest allowed discriminant
3861 : * - maxh: largest allowed class number
3862 : * - L0: norm of class group generator (2, 3, 5, or 7)
3863 : *
3864 : * OUTPUT:
3865 : * - tablelen: length of return value
3866 : *
3867 : * RETURN:
3868 : * - array of {D, h(D), kronecker symbols for small p} */
3869 : static D_entry *
3870 3010 : scanD0(long *tablelen, long *minD, long maxD, long maxh, long L0)
3871 : {
3872 : pari_sp av;
3873 : D_entry *tab;
3874 : long i, lF, cnt;
3875 : GEN F;
3876 :
3877 : /* NB: As seen in the loop below, the real class number of D can be */
3878 : /* 2*maxh if cl(D) is cyclic. */
3879 3010 : tab = (D_entry *) stack_malloc((maxD/4)*sizeof(*tab)); /* Overestimate */
3880 3010 : F = vecfactorsquarefreeu_coprime(*minD, maxD, mkvecsmall(2));
3881 3010 : lF = lg(F);
3882 30084950 : for (av = avma, cnt = 0, i = 1; i < lF; i++, set_avma(av))
3883 : {
3884 30081940 : GEN DD, ordL, f, q = gel(F,i);
3885 : long j, k, n, h, L1, d, D;
3886 : ulong m;
3887 :
3888 30081940 : if (!q) continue; /* not square-free */
3889 : /* restrict to possibly cyclic class groups */
3890 12199514 : k = lg(q) - 1; if (k > 2) continue;
3891 9505100 : d = i + *minD - 1; /* q = prime divisors of d */
3892 9505100 : if ((d & 3) == 1) continue;
3893 4782636 : D = -d; /* d = 3 (mod 4), D = 1 mod 4 fundamental */
3894 4782636 : if (kross(D, L0) < 1) continue;
3895 :
3896 : /* L1 initially the first factor of d if small enough, otherwise ignored */
3897 2299380 : L1 = (k > 1 && q[1] <= MAX_L1)? q[1]: 0;
3898 :
3899 : /* Check if h(D) is too big */
3900 2299380 : h = hclassno6u(d) / 6;
3901 2299380 : if (h > 2*maxh || (!L1 && h > maxh)) continue;
3902 :
3903 : /* Check if ord(f) is not big enough to generate at least half the
3904 : * class group (where f is the L0-primeform). */
3905 2153988 : DD = stoi(D);
3906 2153988 : f = primeform_u(DD, L0);
3907 2153988 : ordL = qfi_order(qfbred_i(f), stoi(h));
3908 2153988 : n = itos(ordL);
3909 2153988 : if (n < h/2 || (!L1 && n < h)) continue;
3910 :
3911 : /* If f is big enough, great! Otherwise, for each potential L1,
3912 : * do a discrete log to see if it is NOT in the subgroup generated
3913 : * by L0; stop as soon as such is found. */
3914 1849564 : for (j = 1;; j++) {
3915 2090880 : if (n == h || (L1 && !qfi_Shanks(primeform_u(DD, L1), f, n))) {
3916 1754830 : dbg_printf(2)("D0=%ld good with L1=%ld\n", D, L1);
3917 1754830 : break;
3918 : }
3919 336050 : if (!L1) break;
3920 241316 : L1 = (j <= k && k > 1 && q[j] <= MAX_L1 ? q[j] : 0);
3921 : }
3922 : /* The first bit of m is set iff f generates a proper subgroup of cl(D)
3923 : * (hence implying that we need L1). */
3924 1849564 : m = (n < h ? 1 : 0);
3925 : /* bits j and j+1 give the 2-bit number 1 + (D|p) where p = prime(j) */
3926 55030912 : for (j = 1 ; j <= SMOOTH_PRIMES; j++)
3927 : {
3928 53181348 : ulong x = (ulong) (1 + kross(D, PRIMES[j]));
3929 53181348 : m |= x << (2*j);
3930 : }
3931 :
3932 : /* Insert d, h and m into the table */
3933 1849564 : tab[cnt].D = D;
3934 1849564 : tab[cnt].h = h;
3935 1849564 : tab[cnt].m = m; cnt++;
3936 : }
3937 :
3938 : /* Sort the table */
3939 3010 : qsort(tab, cnt, sizeof(*tab), _qsort_cmp);
3940 3010 : *tablelen = cnt;
3941 3010 : *minD = maxD + 3 - (maxD & 3); /* smallest d >= maxD, d = 3 (mod 4) */
3942 3010 : return tab;
3943 : }
3944 :
3945 : /* Populate Ds with discriminants (and attached data) that can be
3946 : * used to calculate the modular polynomial of level L and invariant
3947 : * inv. Return the number of discriminants found. */
3948 : static long
3949 3008 : discriminant_with_classno_at_least(disc_info bestD[MODPOLY_MAX_DCNT],
3950 : long L, long inv, GEN Q, long ignore_sparse)
3951 : {
3952 : enum { SMALL_L_BOUND = 101 };
3953 3008 : long max_max_D = 160000 * (inv ? 2 : 1);
3954 : long minD, maxD, maxh, L0, max_L1, minbits, Dcnt, flags, s, d, i, tablen;
3955 : D_entry *tab;
3956 3008 : double eps, cost, best_eps = -1.0, best_cost = -1.0;
3957 : disc_info Ds[MODPOLY_MAX_DCNT];
3958 3008 : long best_cnt = 0;
3959 : pari_timer T;
3960 3008 : timer_start(&T);
3961 :
3962 3008 : s = modinv_sparse_factor(inv);
3963 3008 : d = ceildivuu(L+1, s) + 1;
3964 :
3965 : /* maxD of 10000 allows us to get a satisfactory discriminant in
3966 : * under 250ms in most cases. */
3967 3008 : maxD = 10000;
3968 : /* Allow the class number to overshoot L by 50%. Must be at least
3969 : * 1.1*L, and higher values don't seem to provide much benefit,
3970 : * except when L is small, in which case it's necessary to get any
3971 : * discriminant at all in some cases. */
3972 3008 : maxh = (L / s < SMALL_L_BOUND) ? 10 * L : 1.5 * L;
3973 :
3974 3008 : flags = ignore_sparse ? MODPOLY_IGNORE_SPARSE_FACTOR : 0;
3975 3008 : L0 = select_L0(L, inv, 0);
3976 3008 : max_L1 = L / 2 + 2; /* for L=11 we need L1=7 for j */
3977 3008 : minbits = modpoly_height_bound(L, inv);
3978 3008 : if (Q) minbits += expi(Q);
3979 3008 : minD = 7;
3980 :
3981 6016 : while ( ! best_cnt) {
3982 3010 : while (maxD <= max_max_D) {
3983 : /* TODO: Find a way to re-use tab when we need multiple modpolys */
3984 3010 : tab = scanD0(&tablen, &minD, maxD, maxh, L0);
3985 3010 : dbg_printf(1)("Found %ld potential fundamental discriminants\n", tablen);
3986 :
3987 3010 : Dcnt = modpoly_pickD(Ds, L, inv, L0, max_L1, minbits, flags, tab, tablen);
3988 3010 : eps = 0.0;
3989 3010 : cost = 0.0;
3990 :
3991 3010 : if (Dcnt) {
3992 3008 : long n1 = 0;
3993 6035 : for (i = 0; i < Dcnt; i++) {
3994 3027 : n1 = maxss(n1, Ds[i].n1);
3995 3027 : cost += Ds[i].cost;
3996 : }
3997 3008 : eps = (n1 * s - L) / (double)L;
3998 :
3999 3008 : if (best_cost < 0.0 || cost < best_cost) {
4000 3008 : if (best_cnt)
4001 0 : for (i = 0; i < best_cnt; i++) killblock((GEN)bestD[i].primes);
4002 3008 : (void) memcpy(bestD, Ds, Dcnt * sizeof(disc_info));
4003 3008 : best_cost = cost;
4004 3008 : best_cnt = Dcnt;
4005 3008 : best_eps = eps;
4006 : /* We're satisfied if n1 is within 5% of L. */
4007 3008 : if (L / s <= SMALL_L_BOUND || eps < 0.05) break;
4008 : } else {
4009 0 : for (i = 0; i < Dcnt; i++) killblock((GEN)Ds[i].primes);
4010 : }
4011 : } else {
4012 2 : if (log2(maxD) > BITS_IN_LONG - 2 * (log2(L) + 2))
4013 : {
4014 0 : char *err = stack_sprintf("modular polynomial of level %ld and invariant %ld",L,inv);
4015 0 : pari_err(e_ARCH, err);
4016 : }
4017 : }
4018 2 : maxD *= 2;
4019 2 : minD += 4;
4020 2 : dbg_printf(0)(" Doubling discriminant search space (closest: %.1f%%, cost ratio: %.1f)...\n", eps*100, cost/(double)(d*(L-1)));
4021 : }
4022 3008 : max_max_D *= 2;
4023 : }
4024 :
4025 3008 : if (DEBUGLEVEL > 3) {
4026 0 : pari_sp av = avma;
4027 0 : err_printf("Found discriminant(s):\n");
4028 0 : for (i = 0; i < best_cnt; ++i) {
4029 0 : long h = itos(classno(stoi(bestD[i].D1)));
4030 0 : set_avma(av);
4031 0 : err_printf(" D = %ld, h = %ld, u = %ld, L0 = %ld, L1 = %ld, n1 = %ld, n2 = %ld, cost = %ld\n",
4032 0 : bestD[i].D1, h, usqrt(bestD[i].D1 / bestD[i].D0), bestD[i].L0,
4033 0 : bestD[i].L1, bestD[i].n1, bestD[i].n2, bestD[i].cost);
4034 : }
4035 0 : err_printf("(off target by %.1f%%, cost ratio: %.1f)\n",
4036 0 : best_eps*100, best_cost/(double)(d*(L-1)));
4037 : }
4038 3008 : return best_cnt;
4039 : }
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