Elliptic curves


Elliptic curve structures

An elliptic curve is given by a Weierstrass model

y^2+a_1xy+a_3y = x^3+a_2x^2+a_4x+a_6,

whose discriminant is non-zero. Affine points on E are represented as two-component vectors [x,y]; the point at infinity, i.e. the identity element of the group law, is represented by the one-component vector [0].

Given a vector of coefficients [a_1,a_2,a_3,a_4,a_6], the function ellinit initializes and returns an ell structure. (An additional optional argument allows to specify the base field in case it cannot be inferred from the curve coefficients.) This structure contains data needed by elliptic curve related functions, and is generally passed as a first argument. Expensive data are skipped on initialization: they will be dynamically computed when (and if) needed, and then inserted in the structure. The precise layout of the ell structure is left undefined and should never be used directly. The following member functions are available, depending on the underlying domain.


ellL1(e, {r = 0})

Returns the value at s = 1 of the derivative of order r of the L-function of the elliptic curve e.

  ? e = ellinit("11a1"); \\ order of vanishing is 0
  ? ellL1(e)
  %2 = 0.2538418608559106843377589233
  ? e = ellinit("389a1");  \\ order of vanishing is 2
  ? ellL1(e)
  %4 = -5.384067311837218089235032414 E-29
  ? ellL1(e, 1)
  %5 = 0
  ? ellL1(e, 2)
  %6 = 1.518633000576853540460385214

The main use of this function, after computing at low accuracy the order of vanishing using ellanalyticrank, is to compute the leading term at high accuracy to check (or use) the Birch and Swinnerton-Dyer conjecture:

  ? \p18
    realprecision = 18 significant digits
  ? e = ellinit("5077a1"); ellanalyticrank(e)
  time = 8 ms.
  %1 = [3, 10.3910994007158041]
  ? \p200
    realprecision = 202 significant digits (200 digits displayed)
  ? ellL1(e, 3)
  time = 104 ms.
  %3 = 10.3910994007158041387518505103609170697263563756570092797 [...]

The library syntax is GEN ellL1_bitprec(GEN e, long r, long bitprec).


elladd(E, z1, z2)

Sum of the points z1 and z2 on the elliptic curve corresponding to E.

The library syntax is GEN elladd(GEN E, GEN z1, GEN z2).


ellak(E, n)

Computes the coefficient a_n of the L-function of the elliptic curve E/ℚ, i.e. coefficients of a newform of weight 2 by the modularity theorem (Taniyama-Shimura-Weil conjecture). E must be an ell structure over ℚ as output by ellinit. E must be given by an integral model, not necessarily minimal, although a minimal model will make the function faster.

  ? E = ellinit([0,1]);
  ? ellak(E, 10)
  %2 = 0
  ? e = ellinit([5^4,5^6]); \\ not minimal at 5
  ? ellak(e, 5) \\ wasteful but works
  %3 = -3
  ? E = ellminimalmodel(e); \\ now minimal
  ? ellak(E, 5)
  %5 = -3

If the model is not minimal at a number of bad primes, then the function will be slower on those n divisible by the bad primes. The speed should be comparable for other n:

  ? for(i=1,10^6, ellak(E,5))
  time = 820 ms.
  ? for(i=1,10^6, ellak(e,5)) \\ 5 is bad, markedly slower
  time = 1,249 ms.
  
  ? for(i=1,10^5,ellak(E,5*i))
  time = 977 ms.
  ? for(i=1,10^5,ellak(e,5*i)) \\ still slower but not so much on average
  time = 1,008 ms.

The library syntax is GEN akell(GEN E, GEN n).


ellan(E, n)

Computes the vector of the first n Fourier coefficients a_k corresponding to the elliptic curve E defined over a number field. If E is defined over ℚ, the curve may be given by an arbitrary model, not necessarily minimal, although a minimal model will make the function faster. Over a more general number field, the model must be locally minimal at all primes above 2 and 3.

The library syntax is GEN ellan(GEN E, long n). Also available is GEN ellanQ_zv(GEN e, long n), which returns a t_VECSMALL instead of a t_VEC, saving on memory.


ellanalyticrank(e, {eps})

Returns the order of vanishing at s = 1 of the L-function of the elliptic curve e and the value of the first non-zero derivative. To determine this order, it is assumed that any value less than eps is zero. If eps is omitted, 2-b/2 is used, where b is the current bit precision.

  ? e = ellinit("11a1"); \\ rank 0
  ? ellanalyticrank(e)
  %2 = [0, 0.2538418608559106843377589233]
  ? e = ellinit("37a1"); \\ rank 1
  ? ellanalyticrank(e)
  %4 = [1, 0.3059997738340523018204836835]
  ? e = ellinit("389a1"); \\ rank 2
  ? ellanalyticrank(e)
  %6 = [2, 1.518633000576853540460385214]
  ? e = ellinit("5077a1"); \\ rank 3
  ? ellanalyticrank(e)
  %8 = [3, 10.39109940071580413875185035]

The library syntax is GEN ellanalyticrank_bitprec(GEN e, GEN eps = NULL, long bitprec).


ellap(E, {p})

Let E be an ell structure as output by ellinit, defined over a number field or a finite field 𝔽_q. The argument p is best left omitted if the curve is defined over a finite field, and must be a prime number or a maximal ideal otherwise. This function computes the trace of Frobenius t for the elliptic curve E, defined by the equation #E(𝔽_q) = q+1 - t (for primes of good reduction).

When the characteristic of the finite field is large, the availability of the seadata package will speed the computation.

If the curve is defined over ℚ, p must be explicitly given and the function computes the trace of the reduction over 𝔽_p. The trace of Frobenius is also the a_p coefficient in the curve L-series L(E,s) = ∑_n a_n n-s, whence the function name. The equation must be integral at p but need not be minimal at p; of course, a minimal model will be more efficient.

  ? E = ellinit([0,1]);  \\ y^2 = x^3 + 0.x + 1, defined over Q
  ? ellap(E, 7) \\ 7 necessary here
  %2 = -4       \\ #E(F_7) = 7+1-(-4) = 12
  ? ellcard(E, 7)
  %3 = 12       \\ OK
  
  ? E = ellinit([0,1], 11);  \\ defined over F_11
  ? ellap(E)       \\ no need to repeat 11
  %4 = 0
  ? ellap(E, 11)   \\ ... but it also works
  %5 = 0
  ? ellgroup(E, 13) \\ ouch, inconsistent input!
     ***   at top-level: ellap(E,13)
     ***                 ^-----------
     *** ellap: inconsistent moduli in Rg_to_Fp:
       11
       13
  
  ? Fq = ffgen(ffinit(11,3), 'a); \\ defines F_q := F11^3
  ? E = ellinit([a+1,a], Fq);  \\ y^2 = x^3 + (a+1)x + a, defined over F_q
  ? ellap(E)
  %8 = -3

If the curve is defined over a more general number field than ℚ, the maximal ideal p must be explicitly given in idealprimedec format. If p is above 2 or 3, the function currently assumes (without checking) that the given model is locally minimal at p. There is no restriction at other primes.

  ? K = nfinit(a^2+1); E = ellinit([1+a,0,1,0,0], K);
  ? fa = idealfactor(K, E.disc)
  %2 =
  [  [5, [-2, 1]~, 1, 1, [2, -1; 1, 2]] 1]
  
  [[13, [5, 1]~, 1, 1, [-5, -1; 1, -5]] 2]
  ? ellap(E, fa[1,1])
  %3 = -1 \\ non-split multiplicative reduction
  ? ellap(E, fa[2,1])
  %4 = 1  \\ split multiplicative reduction
  ? P17 = idealprimedec(K,17)[1];
  ? ellap(E, P17)
  %6 = 6  \\ good reduction
  ? E2 = ellchangecurve(E, [17,0,0,0]);
  ? ellap(E2, P17)
  %8 = 6  \\ same, starting from a non-miminal model
  
  ? P3 = idealprimedec(K,3)[1];
  ? E3 = ellchangecurve(E, [3,0,0,0]);
  ? ellap(E, P3)  \\ OK: E is minimal at P3
  %11 = -2
  ? ellap(E3, P3) \\ junk: E3 is not minimal at P3 | 3
  %12 = 0

Algorithms used. If E/𝔽_q has CM by a principal imaginary quadratic order we use a fast explicit formula (involving essentially Kronecker symbols and Cornacchia's algorithm), in O(log q)^2. Otherwise, we use Shanks-Mestre's baby-step/giant-step method, which runs in time Õ(q1/4) using Õ(q1/4) storage, hence becomes unreasonable when q has about 30 digits. Above this range, the SEA algorithm becomes available, heuristically in Õ(log q)^4, and primes of the order of 200 digits become feasible. In small characteristic we use Mestre's (p = 2), Kohel's (p = 3,5,7,13), Satoh-Harley (all in Õ(p2 n^2)) or Kedlaya's (in Õ(p n^3)) algorithms.

The library syntax is GEN ellap(GEN E, GEN p = NULL).


ellbil(E, z1, z2)

Deprecated alias for ellheight(E,P,Q).

The library syntax is GEN bilhell(GEN E, GEN z1, GEN z2, long prec).


ellbsd(E)

The object E being an elliptic curve over a number field, returns a real number c such that the BSD conjecture predicts that LE(r)(1)/r != c R S where r is the rank, R the regulator and S the cardinal of the Tate-Shafarevich group.

  ? e = ellinit([0,-1,1,-10,-20]); \\ rank 0
  ? ellbsd(e)
  %2 = 0.25384186085591068433775892335090946105
  ? lfun(e,1)
  %3 = 0.25384186085591068433775892335090946104
  ? e = ellinit([0,0,1,-1,0]); \\ rank 1
  ? P = ellheegner(e);
  ? ellbsd(e)*ellheight(e,P)
  %6 = 0.30599977383405230182048368332167647445
  ? lfun(e,1,1)
  %7 = 0.30599977383405230182048368332167647445
  ? e = ellinit([1+a,0,1,0,0],nfinit(a^2+1)); \\ rank 0
  ? ellbsd(e)
  %9 = 0.42521832235345764503001271536611593310
  ? lfun(e,1)
  %10 = 0.42521832235345764503001271536611593309

The library syntax is GEN ellbsd(GEN E, long prec).


ellcard(E, {p})

Let E be an ell structure as output by ellinit, defined over ℚ or a finite field 𝔽_q. The argument p is best left omitted if the curve is defined over a finite field, and must be a prime number otherwise. This function computes the order of the group E(𝔽_q) (as would be computed by ellgroup).

When the characteristic of the finite field is large, the availability of the seadata package will speed the computation. See also ellap for the list of implemented algorithms.

If the curve is defined over ℚ, p must be explicitly given and the function computes the cardinality of the reduction over 𝔽_p; the equation need not be minimal at p, but a minimal model will be more efficient. The reduction is allowed to be singular, and we return the order of the group of non-singular points in this case.

The library syntax is GEN ellcard(GEN E, GEN p = NULL). Also available is GEN ellcard(GEN E, GEN p) where p is not NULL.


ellchangecurve(E, v)

Changes the data for the elliptic curve E by changing the coordinates using the vector v = [u,r,s,t], i.e. if x' and y' are the new coordinates, then x = u^2x'+r, y = u^3y'+su^2x'+t. E must be an ell structure as output by ellinit. The special case v = 1 is also used instead of [1,0,0,0] to denote the trivial coordinate change.

The library syntax is GEN ellchangecurve(GEN E, GEN v).


ellchangepoint(x, v)

Changes the coordinates of the point or vector of points x using the vector v = [u,r,s,t], i.e. if x' and y' are the new coordinates, then x = u^2x'+r, y = u^3y'+su^2x'+t (see also ellchangecurve).

  ? E0 = ellinit([1,1]); P0 = [0,1]; v = [1,2,3,4];
  ? E = ellchangecurve(E0, v);
  ? P = ellchangepoint(P0,v)
  %3 = [-2, 3]
  ? ellisoncurve(E, P)
  %4 = 1
  ? ellchangepointinv(P,v)
  %5 = [0, 1]

The library syntax is GEN ellchangepoint(GEN x, GEN v). The reciprocal function GEN ellchangepointinv(GEN x, GEN ch) inverts the coordinate change.


ellchangepointinv(x, v)

Changes the coordinates of the point or vector of points x using the inverse of the isomorphism attached to v = [u,r,s,t], i.e. if x' and y' are the old coordinates, then x = u^2x'+r, y = u^3y'+su^2x'+t (inverse of ellchangepoint).

  ? E0 = ellinit([1,1]); P0 = [0,1]; v = [1,2,3,4];
  ? E = ellchangecurve(E0, v);
  ? P = ellchangepoint(P0,v)
  %3 = [-2, 3]
  ? ellisoncurve(E, P)
  %4 = 1
  ? ellchangepointinv(P,v)
  %5 = [0, 1]  \\ we get back P0

The library syntax is GEN ellchangepointinv(GEN x, GEN v).


ellconvertname(name)

Converts an elliptic curve name, as found in the elldata database, from a string to a triplet [conductor, isogeny class, index]. It will also convert a triplet back to a curve name. Examples:

  ? ellconvertname("123b1")
  %1 = [123, 1, 1]
  ? ellconvertname(%)
  %2 = "123b1"

The library syntax is GEN ellconvertname(GEN name).


elldivpol(E, n, {v = 'x})

n-division polynomial f_n for the curve E in the variable v. In standard notation, for any affine point P = (X,Y) on the curve, we have [n]P = (φ_n(P)ψ_n(P) : ω_n(P) : ψ_n(P)^3) for some polynomials φ_n,ω_n,ψ_n in ℤ[a_1,a_2,a_3,a_4,a_6][X,Y]. We have f_n(X) = ψ_n(X) for n odd, and f_n(X) = ψ_n(X,Y) (2Y + a_1X+a_3) for n even. We have f_1 = 1, f_2 = 4X^3 + b_2X^2 + 2b_4 X + b_6, f_3 = 3 X^4 + b_2 X^3 + 3b_4 X^2 + 3 b_6 X + b8, f_4 = f_2(2X^6 + b_2 X^5 + 5b_4 X^4 + 10 b_6 X^3 + 10 b_8 X^2 + (b_2b_8-b_4b_6)X + (b_8b_4 - b_6^2)),... For n ≥ 2, the roots of f_n are the X-coordinates of points in E[n].

The library syntax is GEN elldivpol(GEN E, long n, long v = -1) where v is a variable number.


elleisnum(w, k, {flag = 0})

k being an even positive integer, computes the numerical value of the Eisenstein series of weight k at the lattice w, as given by ellperiods, namely

(2i π/ω_2)^k (1 + 2/ζ(1-k) ∑n ≥ 1 nk-1q^n / (1-q^n)),

where q = exp(2iπ τ) and τ := ω_1/ω_2 belongs to the complex upper half-plane. It is also possible to directly input w = [ω_1,ω_2], or an elliptic curve E as given by ellinit.

  ? w = ellperiods([1,I]);
  ? elleisnum(w, 4)
  %2 = 2268.8726415508062275167367584190557607
  ? elleisnum(w, 6)
  %3 = -3.977978632282564763 E-33
  ? E = ellinit([1, 0]);
  ? elleisnum(E, 4, 1)
  %5 = -47.999999999999999999999999999999999998

When flag is non-zero and k = 4 or 6, returns the elliptic invariants g_2 or g_3, such that y^2 = 4x^3 - g_2 x - g_3 is a Weierstrass equation for E.

The library syntax is GEN elleisnum(GEN w, long k, long flag, long prec).


elleta(w)

Returns the quasi-periods [η_1,η_2] attached to the lattice basis w = [ω_1, ω_2]. Alternatively, w can be an elliptic curve E as output by ellinit, in which case, the quasi periods attached to the period lattice basis E.omega (namely, E.eta) are returned.

  ? elleta([1, I])
  %1 = [3.141592653589793238462643383, 9.424777960769379715387930149*I]

The library syntax is GEN elleta(GEN w, long prec).


ellformaldifferential(E, {n = seriesprecision}, {t = 'x})

Let ω := dx / (2y+a_1x+a_3) be the invariant differential form attached to the model E of some elliptic curve (ellinit form), and η := x(t)ω. Return n terms (seriesprecision by default) of f(t),g(t) two power series in the formal parameter t = -x/y such that ω = f(t) dt, η = g(t) dt: f(t) = 1+a_1 t + (a_1^2 + a_2) t^2 +..., g(t) = t-2 +...

   ? E = ellinit([-1,1/4]); [f,g] = ellformaldifferential(E,7,'t);
   ? f
   %2 = 1 - 2*t^4 + 3/4*t^6 + O(t^7)
   ? g
   %3 = t^-2 - t^2 + 1/2*t^4 + O(t^5)

The library syntax is GEN ellformaldifferential(GEN E, long precdl, long n = -1) where n is a variable number.


ellformalexp(E, {n = seriesprecision}, {z = 'x})

The elliptic formal exponential Exp attached to E is the isomorphism from the formal additive law to the formal group of E. It is normalized so as to be the inverse of the elliptic logarithm (see ellformallog): Exp o L = Id. Return n terms of this power series:

  ? E=ellinit([-1,1/4]); Exp = ellformalexp(E,10,'z)
  %1 = z + 2/5*z^5 - 3/28*z^7 + 2/15*z^9 + O(z^11)
  ? L = ellformallog(E,10,'t);
  ? subst(Exp,z,L)
  %3 = t + O(t^11)

The library syntax is GEN ellformalexp(GEN E, long precdl, long n = -1) where n is a variable number.


ellformallog(E, {n = seriesprecision}, {v = 'x})

The formal elliptic logarithm is a series L in t K[[t]] such that d L = ω = dx / (2y + a_1x + a_3), the canonical invariant differential attached to the model E. It gives an isomorphism from the formal group of E to the additive formal group.

  ? E = ellinit([-1,1/4]); L = ellformallog(E, 9, 't)
  %1 = t - 2/5*t^5 + 3/28*t^7 + 2/3*t^9 + O(t^10)
  ? [f,g] = ellformaldifferential(E,8,'t);
  ? L' - f
  %3 = O(t^8)

The library syntax is GEN ellformallog(GEN E, long precdl, long n = -1) where n is a variable number.


ellformalpoint(E, {n = seriesprecision}, {v = 'x})

If E is an elliptic curve, return the coordinates x(t), y(t) in the formal group of the elliptic curve E in the formal parameter t = -x/y at oo : x = t-2 -a_1 t-1 - a_2 - a_3 t +... y = - t-3 -a_1 t-2 - a_2t-1 -a_3 +... Return n terms (seriesprecision by default) of these two power series, whose coefficients are in ℤ[a_1,a_2,a_3,a_4,a_6].

  ? E = ellinit([0,0,1,-1,0]); [x,y] = ellformalpoint(E,8,'t);
  ? x
  %2 = t^-2 - t + t^2 - t^4 + 2*t^5 + O(t^6)
  ? y
  %3 = -t^-3 + 1 - t + t^3 - 2*t^4 + O(t^5)
  ? E = ellinit([0,1/2]); ellformalpoint(E,7)
  %4 = [x^-2 - 1/2*x^4 + O(x^5), -x^-3 + 1/2*x^3 + O(x^4)]

The library syntax is GEN ellformalpoint(GEN E, long precdl, long n = -1) where n is a variable number.


ellformalw(E, {n = seriesprecision}, {t = 'x})

Return the formal power series w attached to the elliptic curve E, in the variable t: w(t) = t^3 + a_1 t^4 + (a_2 + a_1^2) t^5 +...+ O(tn+3), which is the formal expansion of -1/y in the formal parameter t := -x/y at oo (take n = seriesprecision if n is omitted). The coefficients of w belong to ℤ[a_1,a_2,a_3,a_4,a_6].

  ? E=ellinit([3,2,-4,-2,5]); ellformalw(E, 5, 't)
  %1 = t^3 + 3*t^4 + 11*t^5 + 35*t^6 + 101*t^7 + O(t^8)

The library syntax is GEN ellformalw(GEN E, long precdl, long n = -1) where n is a variable number.


ellfromeqn(P)

Given a genus 1 plane curve, defined by the affine equation f(x,y) = 0, return the coefficients [a_1,a_2,a_3,a_4,a_6] of a Weierstrass equation for its Jacobian. This allows to recover a Weierstrass model for an elliptic curve given by a general plane cubic or by a binary quartic or biquadratic model. The function implements the f f^* formulae of Artin, Tate and Villegas (Advances in Math. 198 (2005), pp. 366--382).

In the example below, the function is used to convert between twisted Edwards coordinates and Weierstrass coordinates.

  ? e = ellfromeqn(a*x^2+y^2 - (1+d*x^2*y^2))
  %1 = [0, -a - d, 0, -4*d*a, 4*d*a^2 + 4*d^2*a]
  ? E = ellinit(ellfromeqn(y^2-x^2 - 1 +(121665/121666*x^2*y^2)),2^255-19);
  ? isprime(ellcard(E) / 8)
  %3 = 1

The elliptic curve attached to the sum of two cubes is given by

  ? ellfromeqn(x^3+y^3 - a)
  %1 = [0, 0, -9*a, 0, -27*a^2]

Congruent number problem:. Let n be an integer, if a^2+b^2 = c^2 and a b = 2 n, then by substituting b by 2 n/a in the first equation, we get ((a^2+(2 n/a)^2)-c^2) a^2 = 0. We set x = a, y = a c.

  ? En = ellfromeqn((x^2 + (2*n/x)^2 - (y/x)^2)*x^2)
  %1 = [0, 0, 0, -16*n^2, 0]

For example 23 is congruent since the curve has a point of infinite order, namely:

  ? ellheegner( ellinit(subst(En, n, 23)) )
  %2 = [168100/289, 68053440/4913]

The library syntax is GEN ellfromeqn(GEN P).


ellfromj(j)

Returns the coefficients [a_1,a_2,a_3,a_4,a_6] of a fixed elliptic curve with j-invariant j.

The library syntax is GEN ellfromj(GEN j).


ellgenerators(E)

If E is an elliptic curve over the rationals, return a ℤ-basis of the free part of the Mordell-Weil group attached to E. This relies on the elldata database being installed and referencing the curve, and so is only available for curves over ℤ of small conductors. If E is an elliptic curve over a finite field 𝔽_q as output by ellinit, return a minimal set of generators for the group E(𝔽_q).

Caution. when the group is not cyclic, of shape ℤ/d_1ℤ x ℤ/d_2ℤ with d_2 | d_1, the points [P,Q] returned by ellgenerators need not have order d_1 and d_2: it is true that P has order d_1, but we only know that Q is a generator of E(𝔽_q)/ <P> and that the Weil pairing w(P,Q) has order d_2, see ??ellgroup. If you need generators [P,R] with R of order d_2, find x such that R = Q-[x]P has order d_2 by solving the discrete logarithm problem [d_2]Q = [x]([d_2]P) in a cyclic group of order d_1/d_2. This will be very expensive if d_1/d_2 has a large prime factor.

The library syntax is GEN ellgenerators(GEN E).


ellglobalred(E)

Let E be an ell structure as output by ellinit attached to an elliptic curve defined over a number field. This function calculates the arithmetic conductor and the global Tamagawa number c. The result [N,v,c,F,L] is slightly different if E is defined over ℚ (domain D = 1 in ellinit) or over a number field (domain D is a number field structure, including nfinit(x) representing ℚ !):

* N is the arithmetic conductor of the curve,

* v is an obsolete field, left in place for backward compatibility. If E is defined over ℚ, v gives the coordinate change for E to the standard minimal integral model (ellminimalmodel provides it in a cheaper way); if E is defined over another number field, v gives a coordinate change to an integral model (ellintegralmodel provides it in a cheaper way).

* c is the product of the local Tamagawa numbers c_p, a quantity which enters in the Birch and Swinnerton-Dyer conjecture,

* F is the factorization of N,

* L is a vector, whose i-th entry contains the local data at the i-th prime ideal divisor of N, i.e. L[i] = elllocalred(E,F[i,1]). If E is defined over ℚ, the local coordinate change has been deleted and replaced by a 0; if E is defined over another number field the local coordinate change to a local minimal model is given relative to the integral model afforded by v (so either start from an integral model so that v be trivial, or apply v first).

The library syntax is GEN ellglobalred(GEN E).


ellgroup(E, {p}, {flag})

Let E be an ell structure as output by ellinit, defined over ℚ or a finite field 𝔽_q. The argument p is best left omitted if the curve is defined over a finite field, and must be a prime number otherwise. This function computes the structure of the group E(𝔽_q) ~ ℤ/d_1ℤ x ℤ/d_2ℤ, with d_2 | d_1.

If the curve is defined over ℚ, p must be explicitly given and the function computes the structure of the reduction over 𝔽_p; the equation need not be minimal at p, but a minimal model will be more efficient. The reduction is allowed to be singular, and we return the structure of the (cyclic) group of non-singular points in this case.

If the flag is 0 (default), return [d_1] or [d_1, d_2], if d_2 > 1. If the flag is 1, return a triple [h,cyc,gen], where h is the curve cardinality, cyc gives the group structure as a product of cyclic groups (as per flag = 0). More precisely, if d_2 > 1, the output is [d_1d_2, [d_1,d_2],[P,Q]] where P is of order d_1 and [P,Q] generates the curve. Caution. It is not guaranteed that Q has order d_2, which in the worst case requires an expensive discrete log computation. Only that ellweilpairing(E, P, Q, d1) has order d_2.

  ? E = ellinit([0,1]);  \\ y^2 = x^3 + 0.x + 1, defined over Q
  ? ellgroup(E, 7)
  %2 = [6, 2] \\ Z/6 x Z/2, non-cyclic
  ? E = ellinit([0,1] * Mod(1,11));  \\ defined over F_11
  ? ellgroup(E)   \\ no need to repeat 11
  %4 = [12]
  ? ellgroup(E, 11)   \\ ... but it also works
  %5 = [12]
  ? ellgroup(E, 13) \\ ouch, inconsistent input!
     ***   at top-level: ellgroup(E,13)
     ***                 ^--------------
     *** ellgroup: inconsistent moduli in Rg_to_Fp:
       11
       13
  ? ellgroup(E, 7, 1)
  %6 = [12, [6, 2], [[Mod(2, 7), Mod(4, 7)], [Mod(4, 7), Mod(4, 7)]]]

If E is defined over ℚ, we allow singular reduction and in this case we return the structure of the group of non-singular points, satisfying #Ens(𝔽_p) = p - a_p.

  ? E = ellinit([0,5]);
  ? ellgroup(E, 5, 1)
  %2 = [5, [5], [[Mod(4, 5), Mod(2, 5)]]]
  ? ellap(E, 5)
  %3 = 0 \\ additive reduction at 5
  ? E = ellinit([0,-1,0,35,0]);
  ? ellgroup(E, 5, 1)
  %5 = [4, [4], [[Mod(2, 5), Mod(2, 5)]]]
  ? ellap(E, 5)
  %6 = 1 \\ split multiplicative reduction at 5
  ? ellgroup(E, 7, 1)
  %7 = [8, [8], [[Mod(3, 7), Mod(5, 7)]]]
  ? ellap(E, 7)
  %8 = -1 \\ non-split multiplicative reduction at 7

The library syntax is GEN ellgroup0(GEN E, GEN p = NULL, long flag). Also available is GEN ellgroup(GEN E, GEN p), corresponding to flag = 0.


ellheegner(E)

Let E be an elliptic curve over the rationals, assumed to be of (analytic) rank 1. This returns a non-torsion rational point on the curve, whose canonical height is equal to the product of the elliptic regulator by the analytic Sha.

This uses the Heegner point method, described in Cohen GTM 239; the complexity is proportional to the product of the square root of the conductor and the height of the point (thus, it is preferable to apply it to strong Weil curves).

  ? E = ellinit([-157^2,0]);
  ? u = ellheegner(E); print(u[1], "\n", u[2])
  69648970982596494254458225/166136231668185267540804
  538962435089604615078004307258785218335/67716816556077455999228495435742408
  ? ellheegner(ellinit([0,1]))         \\ E has rank 0 !
   ***   at top-level: ellheegner(E=ellinit
   ***                 ^--------------------
   *** ellheegner: The curve has even analytic rank.

The library syntax is GEN ellheegner(GEN E).


ellheight(E, P, {Q})

Global Néron-Tate height h(P) of the point P on the elliptic curve E, defined over ℚ or a number field, using the normalization in Cremona's Algorithms for modular elliptic curves. E must be an ell as output by ellinit; it needs not be given by a minimal model although the computation will be faster if it is.

If the argument Q is present, computes the value of the bilinear form (h(P+Q)-h(P-Q)) / 4.

The library syntax is GEN ellheight0(GEN E, GEN P, GEN Q = NULL, long prec). Also available is GEN ellheight(GEN E, GEN P, long prec) (Q omitted).


ellheightmatrix(E, x)

x being a vector of points, this function outputs the Gram matrix of x with respect to the Néron-Tate height, in other words, the (i,j) component of the matrix is equal to ellbil(E,x[i],x[j]). The rank of this matrix, at least in some approximate sense, gives the rank of the set of points, and if x is a basis of the Mordell-Weil group of E, its determinant is equal to the regulator of E. Note our height normalization follows Cremona's Algorithms for modular elliptic curves: this matrix should be divided by 2 to be in accordance with, e.g., Silverman's normalizations.

The library syntax is GEN ellheightmatrix(GEN E, GEN x, long prec).


ellidentify(E)

Look up the elliptic curve E, defined by an arbitrary model over ℚ, in the elldata database. Return [[N, M, G], C] where N is the curve name in Cremona's elliptic curve database, M is the minimal model, G is a ℤ-basis of the free part of the Mordell-Weil group E(ℚ) and C is the change of coordinates change, suitable for ellchangecurve.

The library syntax is GEN ellidentify(GEN E).


ellinit(x, {D = 1})

Initialize an ell structure, attached to the elliptic curve E. E is either

* a 5-component vector [a_1,a_2,a_3,a_4,a_6] defining the elliptic curve with Weierstrass equation Y^2 + a_1 XY + a_3 Y = X^3 + a_2 X^2 + a_4 X + a_6,

* a 2-component vector [a_4,a_6] defining the elliptic curve with short Weierstrass equation Y^2 = X^3 + a_4 X + a_6,

* a character string in Cremona's notation, e.g. "11a1", in which case the curve is retrieved from the elldata database if available.

The optional argument D describes the domain over which the curve is defined:

* the t_INT 1 (default): the field of rational numbers ℚ.

* a t_INT p, where p is a prime number: the prime finite field 𝔽_p.

* an t_INTMOD Mod(a, p), where p is a prime number: the prime finite field 𝔽_p.

* a t_FFELT, as returned by ffgen: the corresponding finite field 𝔽_q.

* a t_PADIC, O(p^n): the field ℚ_p, where p-adic quantities will be computed to a relative accuracy of n digits. We advise to input a model defined over ℚ for such curves. In any case, if you input an approximate model with t_PADIC coefficients, it will be replaced by a lift to ℚ (an exact model "close" to the one that was input) and all quantities will then be computed in terms of this lifted model, at the given accuracy.

* a t_REAL x: the field ℂ of complex numbers, where floating point quantities are by default computed to a relative accuracy of precision(x). If no such argument is given, the value of realprecision at the time ellinit is called will be used.

* a number field K, given by a nf or bnf structure; a bnf is required for ellminimalmodel.

* a prime ideal 𝔭, given by a prid structure; valid if x is a curve defined over a number field K and the equation is integral and minimal at 𝔭.

This argument D is indicative: the curve coefficients are checked for compatibility, possibly changing D; for instance if D = 1 and an t_INTMOD is found. If inconsistencies are detected, an error is raised:

  ? ellinit([1 + O(5), 1], O(7));
   ***   at top-level: ellinit([1+O(5),1],O
   ***                 ^--------------------
   *** ellinit: inconsistent moduli in ellinit: 7 != 5

If the curve coefficients are too general to fit any of the above domain categories, only basic operations, such as point addition, will be supported later.

If the curve (seen over the domain D) is singular, fail and return an empty vector [].

  ? E = ellinit([0,0,0,0,1]); \\ y^2 = x^3 + 1, over Q
  ? E = ellinit([0,1]);       \\ the same curve, short form
  ? E = ellinit("36a1");      \\ sill the same curve, Cremona's notations
  ? E = ellinit([0,1], 2)     \\ over F2: singular curve
  %4 = []
  ? E = ellinit(['a4,'a6] * Mod(1,5));  \\ over F_5[a4,a6], basic support !

The result of ellinit is an ell structure. It contains at least the following information in its components:

a_1,a_2,a_3,a_4,a_6,b_2,b_4,b_6,b_8,c_4,c_6,Δ,j.

All are accessible via member functions. In particular, the discriminant is E.disc, and the j-invariant is E.j.

  ? E = ellinit([a4, a6]);
  ? E.disc
  %2 = -64*a4^3 - 432*a6^2
  ? E.j
  %3 = -6912*a4^3/(-4*a4^3 - 27*a6^2)

Further components contain domain-specific data, which are in general dynamic: only computed when needed, and then cached in the structure.

  ? E = ellinit([2,3], 10^60+7);  \\ E over F_p, p large
  ? ellap(E)
  time = 4,440 ms.
  %2 = -1376268269510579884904540406082
  ? ellcard(E);  \\ now instantaneous !
  time = 0 ms.
  ? ellgenerators(E);
  time = 5,965 ms.
  ? ellgenerators(E); \\ second time instantaneous
  time = 0 ms.

See the description of member functions related to elliptic curves at the beginning of this section.

The library syntax is GEN ellinit(GEN x, GEN D = NULL, long prec).


ellintegralmodel(E, {&v})

Let E be an ell structure over a number field K. This function returns an integral model. If v is present, sets v = [u,0,0,0] to the corresponding change of variable: the return value is identical to that of ellchangecurve(E, v).

  ? e = ellinit([1/17,1/42]);
  ? e = ellintegralmodel(e,&v);
  ? e[1..5]
  %3 = [0, 0, 0, 15287762448, 3154568630095008]
  ? v
  %4 = [1/714, 0, 0, 0]

The library syntax is GEN ellintegralmodel(GEN E, GEN *v = NULL).


ellisdivisible(E, P, n, {&Q})

Given E/K a number field and P in E(K) return 1 if P = [n]R for some R in E(K) and set Q to one such R; and return 0 otherwise. The integer n ≥ 0 may be given as ellxn(E,n), if many points need to be tested.

  ? K = nfinit(polcyclo(11,t));
  ? E = ellinit([0,-1,1,0,0], K);
  ? P = [0,0];
  ? ellorder(E,P)
  %4 = 5
  ? ellisdivisible(E,P,5, &Q)
  %5 = 1
  ? lift(Q)
  %6 = [-t^7-t^6-t^5-t^4+1, -t^9-2*t^8-2*t^7-3*t^6-3*t^5-2*t^4-2*t^3-t^2-1]
  ? ellorder(E, Q)
  %7 = 25

The algebraic complexity of the underlying algorithm is in O(n^4), so it is advisable to first factor n, then use a chain of checks attached to the prime divisors of n: the function will do it itself unless n is given in ellxn form.

The library syntax is long ellisdivisible(GEN E, GEN P, GEN n, GEN *Q = NULL).


ellisogeny(E, G, {only_image = 0}, {x = 'x}, {y = 'y})

Given an elliptic curve E, a finite subgroup G of E is given either as a generating point P (for a cyclic G) or as a polynomial whose roots vanish on the x-coordinates of the non-zero elements of G (general case and more efficient if available). This function returns the [a_1,a_2,a_3,a_4,a_6] invariants of the quotient elliptic curve E/G and (if only_image is zero (the default)) a vector of rational functions [f, g, h] such that the isogeny E → E/G is given by (x,y) (f(x)/h(x)^2, g(x,y)/h(x)^3).

  ? E = ellinit([0,1]);
  ? elltors(E)
  %2 = [6, [6], [[2, 3]]]
  ? ellisogeny(E, [2,3], 1)  \\ Weierstrass model for E/<P>
  %3 = [0, 0, 0, -135, -594]
  ? ellisogeny(E,[-1,0])
  %4 = [[0,0,0,-15,22], [x^3+2*x^2+4*x+3, y*x^3+3*y*x^2-2*y, x+1]]

The library syntax is GEN ellisogeny(GEN E, GEN G, long only_image, long x = -1, long y = -1) where x, y are variable numbers.


ellisogenyapply(f, g)

Given an isogeny of elliptic curves f:E' → E (being the result of a call to ellisogeny), apply f to g:

* if g is a point P in the domain of f, return the image f(P);

* if g:E" → E' is a compatible isogeny, return the composite isogeny f o g: E" → E.

  ? one = ffgen(101, 't)^0;
  ? E = ellinit([6, 53, 85, 32, 34] * one);
  ? P = [84, 71] * one;
  ? ellorder(E, P)
  %4 = 5
  ? [F, f] = ellisogeny(E, P);  \\ f: E->F = E/<P>
  ? ellisogenyapply(f, P)
  %6 = [0]
  ? F = ellinit(F);
  ? Q = [89, 44] * one;
  ? ellorder(F, Q)
  %9 = 2
  ? [G, g] = ellisogeny(F, Q); \\  g: F->G = F/<Q>
  ? gof = ellisogenyapply(g, f); \\ gof: E -> G

The library syntax is GEN ellisogenyapply(GEN f, GEN g).


ellisomat(E, {p = 0}, {fl = 0})

Given an elliptic curve E defined over a number field, compute representatives of the isomorphism classes of elliptic curves ℚ-isogenous to E. The function returns a vector [L,M] where L is a list of triples [E_i, f_i, g_i], where E_i is an elliptic curve in [a_4,a_6] form, f_i: E → E_i is a rational isogeny, g_i: E_i → E is the dual isogeny of f_i, and M is the matrix such that Mi,j is the degree of the isogeny between E_i and E_j. Furthermore the first curve E_1 is isomorphic to E by f_1. If the flag fl = 1, the f_i and g_i are not computed, which saves time, and L is the list of the curves E_i. If p is set it must be a prime number, in which case only isogenies of degree a power of p are considered. Currently the function has some limitation over a number field: p must be set and the curve must not have CM.

  ? E = ellinit("14a1");
  ? [L,M] = ellisomat(E);
  ? LE = apply(x->x[1], L)  \\ list of curves
  %3 = [[215/48,-5291/864],[-675/16,6831/32],[-8185/48,-742643/864],
       [-1705/48,-57707/864],[-13635/16,306207/32],[-131065/48,-47449331/864]]
  ? L[2][2]  \\ isogeny f_2
  %4 = [x^3+3/4*x^2+19/2*x-311/12,
        1/2*x^4+(y+1)*x^3+(y-4)*x^2+(-9*y+23)*x+(55*y+55/2),x+1/3]
  ? L[2][3]  \\ dual isogeny g_2
  %5 = [1/9*x^3-1/4*x^2-141/16*x+5613/64,
        -1/18*x^4+(1/27*y-1/3)*x^3+(-1/12*y+87/16)*x^2+(49/16*y-48)*x
        +(-3601/64*y+16947/512),x-3/4]
  ? apply(E->ellidentify(ellinit(E))[1][1], LE)
  %6 = ["14a1","14a4","14a3","14a2","14a6","14a5"]
  ? M
  %7 =
  [1  3  3 2  6  6]
  
  [3  1  9 6  2 18]
  
  [3  9  1 6 18  2]
  
  [2  6  6 1  3  3]
  
  [6  2 18 3  1  9]
  
  [6 18  2 3  9  1]

The library syntax is GEN ellisomat(GEN E, long p, long fl).


ellisoncurve(E, z)

Gives 1 (i.e. true) if the point z is on the elliptic curve E, 0 otherwise. If E or z have imprecise coefficients, an attempt is made to take this into account, i.e. an imprecise equality is checked, not a precise one. It is allowed for z to be a vector of points in which case a vector (of the same type) is returned.

The library syntax is GEN ellisoncurve(GEN E, GEN z). Also available is int oncurve(GEN E, GEN z) which does not accept vectors of points.


ellissupersingular(E, {p})

Return 1 if the elliptic curve E defined over a number field or a finite field is supersingular at p, and 0 otherwise. If the curve is defined over a number field, p must be explicitly given, and must be a prime number, resp. a maximal ideal, if the curve is defined over ℚ, resp. a general number field: we return 1 if and only if E has supersingular good reduction at p.

Alternatively, E can be given by its j-invariant in a finite field. In this case p must be omitted.

  ? setrand(1); \\ make the choice of g deterministic
  ? g = ffprimroot(ffgen(7^5))
  %1 = 4*x^4 + 5*x^3 + 6*x^2 + 5*x + 6
  ? [g^n | n <- [1 .. 7^5 - 1], ellissupersingular(g^n)]
  %2 = [6]
  
  ? K = nfinit(y^3-2); P = idealprimedec(K, 2)[1];
  ? E = ellinit([y,1], K);
  ? ellissupersingular(E, P)
  %5 = 1
  ? Q = idealprimedec(K,5)[1];
  ? ellissupersingular(E, Q)
  %6 = 0

The library syntax is GEN ellissupersingular(GEN E, GEN p = NULL). Also available is int elljissupersingular(GEN j) where j is a j-invariant of a curve over a finite field.


ellj(x)

Elliptic j-invariant. x must be a complex number with positive imaginary part, or convertible into a power series or a p-adic number with positive valuation.

The library syntax is GEN jell(GEN x, long prec).


elllocalred(E, p)

Calculates the Kodaira type of the local fiber of the elliptic curve E at p. E must be an ell structure as output by ellinit, over ℚ (p a rational prime) or a number field K (p a maximal ideal given by a prid structure), and is assumed to have all its coefficients a_i integral. The result is a 4-component vector [f,kod,v,c]. Here f is the exponent of p in the arithmetic conductor of E, and kod is the Kodaira type which is coded as follows:

1 means good reduction (type I_0), 2, 3 and 4 mean types II, III and IV respectively, 4+ν with ν > 0 means type I_ν; finally the opposite values -1, -2, etc. refer to the starred types I_0^*, II^*, etc. The third component v is itself a vector [u,r,s,t] giving the coordinate changes done during the local reduction; u = 1 if and only if the given equation was already minimal at p. Finally, the last component c is the local Tamagawa number c_p.

The library syntax is GEN elllocalred(GEN E, GEN p).


elllog(E, P, G, {o})

Given two points P and G on the elliptic curve E/𝔽_q, returns the discrete logarithm of P in base G, i.e. the smallest non-negative integer n such that P = [n]G. See znlog for the limitations of the underlying discrete log algorithms. If present, o represents the order of G, see Section se:DLfun; the preferred format for this parameter is [N, factor(N)], where N is the order of G.

If no o is given, assume that G generates the curve. The function also assumes that P is a multiple of G.

  ? a = ffgen(ffinit(2,8),'a);
  ? E = ellinit([a,1,0,0,1]);  \\ over F2^8
  ? x = a^3; y = ellordinate(E,x)[1];
  ? P = [x,y]; G = ellmul(E, P, 113);
  ? ord = [242, factor(242)]; \\ P generates a group of order 242. Initialize.
  ? ellorder(E, G, ord)
  %4 = 242
  ? e = elllog(E, P, G, ord)
  %5 = 15
  ? ellmul(E,G,e) == P
  %6 = 1

The library syntax is GEN elllog(GEN E, GEN P, GEN G, GEN o = NULL).


elllseries(E, s, {A = 1})

This function is deprecated, use lfun(E,s) instead.

E being an elliptic curve, given by an arbitrary model over ℚ as output by ellinit, this function computes the value of the L-series of E at the (complex) point s. This function uses an O(N1/2) algorithm, where N is the conductor.

The optional parameter A fixes a cutoff point for the integral and is best left omitted; the result must be independent of A, up to realprecision, so this allows to check the function's accuracy.

The library syntax is GEN elllseries(GEN E, GEN s, GEN A = NULL, long prec).


ellminimaldisc(E)

E being an elliptic curve defined over a number field output by ellinit, return the minimal discriminant ideal of E.

The library syntax is GEN ellminimaldisc(GEN E).


ellminimalmodel(E, {&v})

Let E be an ell structure over a number field K. This function determines whether E admits a global minimal integral model. If so, it returns it and sets v = [u,r,s,t] to the corresponding change of variable: the return value is identical to that of ellchangecurve(E, v).

Else return the (non-principal) Weierstrass class of E, i.e. the class of ∏ 𝔭(v𝔭 - δ𝔭) / 12} where Δ = E.disc is the model's discriminant and 𝔭 ^ δ𝔭 is the local minimal discriminant. This function requires either that E be defined over the rational field ℚ (with domain D = 1 in ellinit), in which case a global minimal model always exists, or over a number field given by a bnf structure. The Weierstrass class is given in bnfisprincipal format, i.e. in terms of the K.gen generators.

The resulting model has integral coefficients and is everywhere minimal, the coefficients a_1 and a_3 are reduced modulo 2 (in terms of the fixed integral basis K.zk) and a_2 is reduced modulo 3. Over ℚ, we further require that a_1 and a_3 be 0 or 1, that a_2 be 0 or ± 1 and that u > 0 in the change of variable: both the model and the change of variable v are then unique.

  ? e = ellinit([6,6,12,55,233]);  \\ over Q
  ? E = ellminimalmodel(e, &v);
  ? E[1..5]
  %3 = [0, 0, 0, 1, 1]
  ? v
  %4 = [2, -5, -3, 9]

  ? K = bnfinit(a^2-65);  \\ over a non-principal number field
  ? K.cyc
  %2 = [2]
  ? u = Mod(8+a, K.pol);
  ? E = ellinit([1,40*u+1,0,25*u^2,0], K);
  ? ellminimalmodel(E) \\ no global minimal model exists over Z_K
  %6 = [1]~

The library syntax is GEN ellminimalmodel(GEN E, GEN *v = NULL).


ellminimaltwist(E, {flag = 0})

Let E be an elliptic curve defined over ℚ, return a discriminant D such that the twist of E by D is minimal among all possible quadratic twists, i.e. if flag = 0, its minimal model has minimal discriminant, or if flag = 1, it has minimal conductor.

In the example below, we find a curve with j-invariant 3 and minimal conductor.

  ? E = ellminimalmodel(ellinit(ellfromj(3)));
  ? ellglobalred(E)[1]
  %2 = 357075
  ? D = ellminimaltwist(E,1)
  %3 = -15
  ? E2 = ellminimalmodel(ellinit(elltwist(E,D)));
  ? ellglobalred(E2)[1]
  %5 = 14283

In the example below, flag = 0 and flag = 1 give different results.

  ? E = ellinit([1,0]);
  ? D0 = ellminimaltwist(E,0)
  %7 = 1
  ? D1 = ellminimaltwist(E,1)
  %8 = 8
  ? E0 = ellminimalmodel(ellinit(elltwist(E,D0)));
  ? [E0.disc, ellglobalred(E0)[1]]
  %10 = [-64, 64]
  ? E1 = ellminimalmodel(ellinit(elltwist(E,D1)));
  ? [E1.disc, ellglobalred(E1)[1]]
  %12 = [-4096, 32]

The library syntax is GEN ellminimaltwist0(GEN E, long flag). Also available are GEN ellminimaltwist(E) for flag = 0, and GEN ellminimaltwistcond(E) for flag = 1.


ellmoddegree(e)

e being an elliptic curve defined over ℚ output by ellinit, compute the modular degree of e divided by the square of the Manin constant. Return [D, err], where D is a rational number and err is exponent of the truncation error.

The library syntax is GEN ellmoddegree(GEN e, long bitprec).


ellmodulareqn(N, {x}, {y})

Given a prime N < 500, return a vector [P,t] where P(x,y) is a modular equation of level N, i.e. a bivariate polynomial with integer coefficients; t indicates the type of this equation: either canonical (t = 0) or Atkin (t = 1). This function requires the seadata package and its only use is to give access to the package contents. See polmodular for a more general and more flexible function.

Let j be the j-invariant function. The polynomial P satisfies the functional equation, P(f,j) = P(f | W_N, j | W_N) = 0 for some modular function f = f_N (hand-picked for each fixed N to minimize its size, see below), where W_N(τ) = -1 / (N τ) is the Atkin-Lehner involution. These two equations allow to compute the values of the classical modular polynomial Φ_N, such that Φ_N(j(τ), j(Nτ)) = 0, while being much smaller than the latter. More precisely, we have j(W_N(τ)) = j(N τ); the function f is invariant under Γ_0(N) and also satisfies

* for Atkin type: f | W_N = f;

* for canonical type: let s = 12/gcd(12,N-1), then f | W_N = N^s / f. In this case, f has a simple definition: f(τ) = N^s (η(N τ) / η(τ) )2 s, where η is Dedekind's eta function.

The following GP function returns values of the classical modular polynomial by eliminating f_N(τ) in the above functional equation, for N ≤ 31 or N ∈ {41,47,59,71}.

  classicaleqn(N, X='X, Y='Y)=
  {
    my([P,t] = ellmodulareqn(N), Q, d);
    if (poldegree(P,'y) > 2, error("level unavailable in classicaleqn"));
    if (t == 0, \\ Canonical
      my(s = 12/gcd(12,N-1));
      Q = 'x^(N+1) * substvec(P,['x,'y],[N^s/'x,Y]);
      d = N^(s*(2*N+1)) * (-1)^(N+1);
    , \\ Atkin
      Q = subst(P,'y,Y);
      d = (X-Y)^(N+1));
    polresultant(subst(P,'y,X), Q) / d;
  }

The library syntax is GEN ellmodulareqn(long N, long x = -1, long y = -1) where x, y are variable numbers.


ellmul(E, z, n)

Computes [n]z, where z is a point on the elliptic curve E. The exponent n is in ℤ, or may be a complex quadratic integer if the curve E has complex multiplication by n (if not, an error message is issued).

  ? Ei = ellinit([1,0]); z = [0,0];
  ? ellmul(Ei, z, 10)
  %2 = [0]     \\ unsurprising: z has order 2
  ? ellmul(Ei, z, I)
  %3 = [0, 0]  \\ Ei has complex multiplication by Z[i]
  ? ellmul(Ei, z, quadgen(-4))
  %4 = [0, 0]  \\ an alternative syntax for the same query
  ? Ej  = ellinit([0,1]); z = [-1,0];
  ? ellmul(Ej, z, I)
    ***   at top-level: ellmul(Ej,z,I)
    ***                 ^--------------
    *** ellmul: not a complex multiplication in ellmul.
  ? ellmul(Ej, z, 1+quadgen(-3))
  %6 = [1 - w, 0]

The simple-minded algorithm for the CM case assumes that we are in characteristic 0, and that the quadratic order to which n belongs has small discriminant.

The library syntax is GEN ellmul(GEN E, GEN z, GEN n).


ellneg(E, z)

Opposite of the point z on elliptic curve E.

The library syntax is GEN ellneg(GEN E, GEN z).


ellnonsingularmultiple(E, P)

Given an elliptic curve E/ℚ (more precisely, a model defined over ℚ of a curve) and a rational point P ∈ E(ℚ), returns the pair [R,n], where n is the least positive integer such that R := [n]P has good reduction at every prime. More precisely, its image in a minimal model is everywhere non-singular.

  ? e = ellinit("57a1"); P = [2,-2];
  ? ellnonsingularmultiple(e, P)
  %2 = [[1, -1], 2]
  ? e = ellinit("396b2"); P = [35, -198];
  ? [R,n] = ellnonsingularmultiple(e, P);
  ? n
  %5 = 12

The library syntax is GEN ellnonsingularmultiple(GEN E, GEN P).


ellorder(E, z, {o})

Gives the order of the point z on the elliptic curve E, defined over a finite field or a number field. Return (the impossible value) zero if the point has infinite order.

  ? E = ellinit([-157^2,0]);  \\ the "157-is-congruent" curve
  ? P = [2,2]; ellorder(E, P)
  %2 = 2
  ? P = ellheegner(E); ellorder(E, P) \\ infinite order
  %3 = 0
  ? K = nfinit(polcyclo(11,t)); E=ellinit("11a3", K); T = elltors(E);
  ? ellorder(E, T.gen[1])
  %5 = 25
  ? E = ellinit(ellfromj(ffgen(5^10)));
  ? ellcard(E)
  %7 = 9762580
  ? P = random(E); ellorder(E, P)
  %8 = 4881290
  ? p = 2^160+7; E = ellinit([1,2], p);
  ? N = ellcard(E)
  %9 = 1461501637330902918203686560289225285992592471152
  ? o = [N, factor(N)];
  ? for(i=1,100, ellorder(E,random(E)))
  time = 260 ms.

The parameter o, is now mostly useless, and kept for backward compatibility. If present, it represents a non-zero multiple of the order of z, see Section se:DLfun; the preferred format for this parameter is [ord, factor(ord)], where ord is the cardinality of the curve. It is no longer needed since PARI is now able to compute it over large finite fields (was restricted to small prime fields at the time this feature was introduced), and caches the result in E so that it is computed and factored only once. Modifying the last example, we see that including this extra parameter provides no improvement:

  ? o = [N, factor(N)];
  ? for(i=1,100, ellorder(E,random(E),o))
  time = 260 ms.

The library syntax is GEN ellorder(GEN E, GEN z, GEN o = NULL). The obsolete form GEN orderell(GEN e, GEN z) should no longer be used.


ellordinate(E, x)

Gives a 0, 1 or 2-component vector containing the y-coordinates of the points of the curve E having x as x-coordinate.

The library syntax is GEN ellordinate(GEN E, GEN x, long prec).


ellpadicL(E, p, n, {s = 0}, {r = 0}, {D = 1})

Returns the value (or r-th derivative) on a character χ^s of ℤ_p^* of the p-adic L-function of the elliptic curve E/ℚ, twisted by D, given modulo p^n.

Characters. The set of continuous characters of Gal(ℚ(μp oo )/ ℚ) is identified to ℤ_p^* via the cyclotomic character χ with values in ℚ_p^*. Denote by τ:ℤ_p^* → ℤ_p^* the Teichmüller character, with values in the (p-1)-th roots of 1 for p != 2, and {-1,1} for p = 2; finally, let <χ>= χ τ-1, with values in 1 + 2pℤ_p. In GP, the continuous character of Gal(ℚ(μp oo )/ ℚ) given by <χ>s_1 τs_2 is represented by the pair of integers s = (s_1,s_2), with s_1 ∈ ℤ_p and s_2 mod p-1 for p > 2, (resp. mod 2 for p = 2); s may be also an integer, representing (s,s) or χ^s.

The p-adic L function. The p-adic L function L_p is defined on the set of continuous characters of Gal(ℚ(μp oo )/ ℚ), as ∫ℤ_p^* χ^s d μ for a certain p-adic distribution μ on ℤ_p^*. The derivative is given by L_p(r)(E, χ^s) = ∫ℤ_p^* log_p^r(a) χ^s(a) dμ(a). More precisely:

* When E has good supersingular reduction, L_p takes its values in ℚ_p ⨂ H^1dR(E/ℚ) and satisfies (1-p-1 F)-2 L_p(E, χ^0) = (L(E,1) / Ω).ω where F is the Frobenius, L(E,1) is the value of the complex L function at 1, ω is the Néron differential and Ω the attached period on E(ℝ). Here, χ^0 represents the trivial character.

The function returns the components of L_p(r)(E,χ^s) in the basis (ω, F(ω)).

* When E has ordinary good reduction, this method only defines the projection of L_p(E,χ^s) on the α-eigenspace, where α is the unit eigenvalue for F. This is what the function returns. We have (1- α-1)-2 Lp,α(E,χ^0) = L(E,1) / Ω.

Two supersingular examples:

  ? cxL(e) = bestappr( ellL1(e) / e.omega[1] );
  
  ? e = ellinit("17a1"); p=3; \\ supersingular, a3 = 0
  ? L = ellpadicL(e,p,4);
  ? F = [0,-p;1,ellap(e,p)]; \\ Frobenius matrix in the basis (omega,F(omega))
  ? (1-p^(-1)*F)^-2 * L / cxL(e)
  %5 = [1 + O(3^5), O(3^5)]~ \\ [1,0]~
  
  ? e = ellinit("116a1"); p=3; \\ supersingular, a3 != 0~
  ? L = ellpadicL(e,p,4);
  ? F = [0,-p; 1,ellap(e,p)];
  ? (1-p^(-1)*F)^-2*L~ / cxL(e)
  %9 = [1 + O(3^4), O(3^5)]~

Good ordinary reduction:

  ? e = ellinit("17a1"); p=5; ap = ellap(e,p)
  %1 = -2 \\ ordinary
  ? L = ellpadicL(e,p,4)
  %2 = 4 + 3*5 + 4*5^2 + 2*5^3 + O(5^4)
  ? al = padicappr(x^2 - ap*x + p, ap + O(p^7))[1];
  ? (1-al^(-1))^(-2) * L / cxL(e)
  %4 = 1 + O(5^4)

Twist and Teichmüller:

  ? e = ellinit("17a1"); p=5; \\ ordinary
  \\ 2nd derivative at tau^1, twist by -7
  ? ellpadicL(e, p, 4, [0,1], 2, -7)
  %2 = 2*5^2 + 5^3 + O(5^4)

This function is a special case of mspadicL, and it also appears as the first term of mspadicseries:

  ? e = ellinit("17a1"); p=5;
  ? L = ellpadicL(e,p,4)
  %2 = 4 + 3*5 + 4*5^2 + 2*5^3 + O(5^4)
  ? [M,phi] = msfromell(e, 1);
  ? Mp = mspadicinit(M, p, 4);
  ? mu = mspadicmoments(Mp, phi);
  ? mspadicL(mu)
  %6 = 4 + 3*5 + 4*5^2 + 2*5^3 + 2*5^4 + 5^5 + O(5^6)
  ? mspadicseries(mu)
  %7 = (4 + 3*5 + 4*5^2 + 2*5^3 + 2*5^4 + 5^5 + O(5^6))
        + (3 + 3*5 + 5^2 + 5^3 + O(5^4))*x
        + (2 + 3*5 + 5^2 + O(5^3))*x^2
        + (3 + 4*5 + 4*5^2 + O(5^3))*x^3
        + (3 + 2*5 + O(5^2))*x^4 + O(x^5)

These are more cumbersome than ellpadicL but allow to compute at different characters, or successive derivatives, or to twist by a quadratic character essentially for the cost of a single call to ellpadicL due to precomputations.

The library syntax is GEN ellpadicL(GEN E, GEN p, long n, GEN s = NULL, long r, GEN D = NULL).


ellpadicfrobenius(E, p, n)

If p > 2 is a prime and E is an elliptic curve on ℚ with good reduction at p, return the matrix of the Frobenius endomorphism ϕ on the crystalline module D_p(E) = ℚ_p ⨂ H^1dR(E/ℚ) with respect to the basis of the given model (ω, η = x ω), where ω = dx/(2 y+a_1 x+a_3) is the invariant differential. The characteristic polynomial of ϕ is x^2 - a_p x + p. The matrix is computed to absolute p-adic precision p^n.

  ? E = ellinit([1,-1,1,0,0]);
  ? F = ellpadicfrobenius(E,5,3);
  ? lift(F)
  %3 =
  [120 29]
  
  [ 55  5]
  ? charpoly(F)
  %4 = x^2 + O(5^3)*x + (5 + O(5^3))
  ? ellap(E, 5)
  %5 = 0

The library syntax is GEN ellpadicfrobenius(GEN E, long p, long n).


ellpadicheight(E, p, n, P, {Q})

Cyclotomic p-adic height of the rational point P on the elliptic curve E (defined over ℚ), given to n p-adic digits. If the argument Q is present, computes the value of the bilinear form (h(P+Q)-h(P-Q)) / 4.

Let DdR(E) := H^1dR(E) ⨂ _ℚ ℚ_p be the ℚ_p vector space spanned by ω (invariant differential dx/(2y+a_1x+a3) related to the given model) and η = x ω. Then the cyclotomic p-adic height associates to P ∈ E(ℚ) an element f ω + gη in DdR. This routine returns the vector [f, g] to n p-adic digits.

If P ∈ E(ℚ) is in the kernel of reduction mod p and if its reduction at all finite places is non singular, then g = -(log_E P)^2, where log_E is the logarithm for the formal group of E at p.

If furthermore the model is of the form Y^2 = X^3 + a X + b and P = (x,y), then f = log_p(denominator(x)) - 2 log_p(σ(P)) where σ(P) is given by ellsigma(E,P).

Recall (Advanced topics in the arithmetic of elliptic curves, Theorem 3.2) that the local height function over the complex numbers is of the form λ(z) = -log (|E.disc|) / 6 + Re(z η(z)) - 2 log( σ(z)). (N.B. our normalization for local and global heights is twice that of Silverman's).

   ? E = ellinit([1,-1,1,0,0]); P = [0,0];
   ? ellpadicheight(E,5,3, P)
   %2 = [3*5 + 5^2 + 2*5^3 + O(5^4), 5^2 + 4*5^4 + O(5^5)]
   ? E = ellinit("11a1"); P = [5,5]; \\ torsion point
   ? ellpadicheight(E,19,6, P)
   %4 = [0, 0]
   ? E = ellinit([0,0,1,-4,2]); P = [-2,1];
   ? ellpadicheight(E,3,3, P)
   %6 = [2*3^2 + 2*3^3 + 3^4 + O(3^5), 2*3^2 + 3^4 + O(3^5)]
   ? ellpadicheight(E,3,5, P, elladd(E,P,P))
   %7 = [3^2 + 2*3^3 + O(3^7), 3^2 + 3^3 + 2*3^4 + 3^5 + O(3^7)]

One can replace the parameter p (prime) by a vector [p,[a,b]], in which case the routine returns the p-adic number af + bg.

When E has good ordinary reduction at p, the "canonical" p-adic height is given by

  s2 = ellpadics2(E,p,n);
  ellpadicheight(E, [p,[1,-s2]], n, P)

Since s_2 does not depend on P, it is preferable to compute it only once:

  ? E = ellinit("5077a1"); p = 5; n = 7;
  ? s2 = ellpadics2(E,p,n);
  ? M = ellpadicheightmatrix(E,[p,[1,-s2]], n, E.gen);
  ? matdet(M)   \\ p-adic regulator
  %4 = 5 + 5^2 + 4*5^3 + 2*5^4 + 2*5^5 + 2*5^6 + O(5^7)

The library syntax is GEN ellpadicheight0(GEN E, GEN p, long n, GEN P, GEN Q = NULL).


ellpadicheightmatrix(E, p, n, v)

v being a vector of points, this function outputs the Gram matrix of v with respect to the cyclotomic p-adic height, given to n p-adic digits; in other words, the (i,j) component of the matrix is equal to ellpadicheight(E,p,n, v[i],v[j]) = [f,g].

See ellpadicheight; in particular one can replace the parameter p prime by a vector [p,[a,b]], in which case the routine returns the matrix containing the p-adic numbers af + bg.

  ? E = ellinit([0,0,1,-7,6]);
  ? ellpadicheightmatrix(E,5,1,[[-2,3],[-1,3]])
  %31 =
  [[4 + O(5), 4*5^2 + O(5^3)]     [O(5), 4*5^2 + O(5^3)]]
  
  [    [O(5), 4*5^2 + O(5^3)] [4 + O(5), 4*5^2 + O(5^3)]]

The library syntax is GEN ellpadicheightmatrix(GEN E, GEN p, long n, GEN v).


ellpadiclog(E, p, n, P)

Given E defined over K = ℚ or ℚ_p and P = [x,y] on E(K) in the kernel of reduction mod p, let t(P) = -x/y be the formal group parameter; this function returns L(t), where L denotes the formal logarithm (mapping the formal group of E to the additive formal group) attached to the canonical invariant differential: dL = dx/(2y + a_1x + a_3).

  ? E = ellinit([0,0,1,-4,2]); P = [-2,1];
  ? ellpadiclog(E,2,10,P)
  %2 = 2 + 2^3 + 2^8 + 2^9 + 2^10 + O(2^11)
  ? E = ellinit([17,42]);
  ? p=3; Ep = ellinit(E,p); \\ E mod p
  ? P=[114,1218]; ellorder(Ep,P) \\ the order of P on (E mod p) is 2
  %5 = 2
  ? Q = ellmul(E,P,2) \\ we need a point of the form 2*P
  %6 = [200257/7056, 90637343/592704]
  ? ellpadiclog(E,3,10,Q)
  %7 = 3 + 2*3^2 + 3^3 + 3^4 + 3^5 + 3^6 + 2*3^8 + 3^9 + 2*3^10 + O(3^11)

The library syntax is GEN ellpadiclog(GEN E, GEN p, long n, GEN P).


ellpadics2(E, p, n)

If p > 2 is a prime and E/ℚ is an elliptic curve with ordinary good reduction at p, returns the slope of the unit eigenvector of ellpadicfrobenius(E,p,n), i.e., the action of Frobenius ϕ on the crystalline module D_p(E) = ℚ_p ⨂ H^1dR(E/ℚ) in the basis of the given model (ω, η = x ω), where ω is the invariant differential dx/(2 y+a_1 x+a_3). In other words, η + s_2ω is an eigenvector for the unit eigenvalue of ϕ.

  ? e=ellinit([17,42]);
  ? ellpadics2(e,13,4)
  %2 = 10 + 2*13 + 6*13^3 + O(13^4)

This slope is the unique c ∈ 3-1ℤ_p such that the odd solution σ(t) = t + O(t^2) of - d((1)/(σ) (d σ)/(ω)) = (x(t) + c) ω is in tℤ_p[[t]].

It is equal to b_2/12 - E_2/12 where E_2 is the value of the Katz p-adic Eisenstein series of weight 2 on (E,ω). This is used to construct a canonical p-adic height when E has good ordinary reduction at p as follows

  s2 = ellpadics2(E,p,n);
  h(E,p,n, P, s2) = ellpadicheight(E, [p,[1,-s2]],n, P);

Since s_2 does not depend on the point P, we compute it only once.

The library syntax is GEN ellpadics2(GEN E, GEN p, long n).


ellperiods(w, {flag = 0})

Let w describe a complex period lattice (w = [w_1,w_2] or an ellinit structure). Returns normalized periods [W_1,W_2] generating the same lattice such that τ := W_1/W_2 has positive imaginary part and lies in the standard fundamental domain for SL_2(ℤ).

If flag = 1, the function returns [[W_1,W_2], [η_1,η_2]], where η_1 and η_2 are the quasi-periods attached to [W_1,W_2], satisfying η_1 W_2 - η_2 W_1 = 2 i π.

The output of this function is meant to be used as the first argument given to ellwp, ellzeta, ellsigma or elleisnum. Quasi-periods are needed by ellzeta and ellsigma only.

The library syntax is GEN ellperiods(GEN w, long flag, long prec).


ellpointtoz(E, P)

If E/ℂ ~ ℂ/Λ is a complex elliptic curve (Λ = E.omega), computes a complex number z, well-defined modulo the lattice Λ, corresponding to the point P; i.e. such that P = [℘_Λ(z),℘'_Λ(z)] satisfies the equation y^2 = 4x^3 - g_2 x - g_3, where g_2, g_3 are the elliptic invariants.

If E is defined over ℝ and P ∈ E(ℝ), we have more precisely, 0 ≤ Re(t) < w1 and 0 ≤ Im(t) < Im(w2), where (w1,w2) are the real and complex periods of E.

  ? E = ellinit([0,1]); P = [2,3];
  ? z = ellpointtoz(E, P)
  %2 = 3.5054552633136356529375476976257353387
  ? ellwp(E, z)
  %3 = 2.0000000000000000000000000000000000000
  ? ellztopoint(E, z) - P
  %4 = [2.548947057811923643 E-57, 7.646841173435770930 E-57]
  ? ellpointtoz(E, [0]) \\ the point at infinity
  %5 = 0

If E/ℚ_p has multiplicative reduction, then E/ℚ_p is analytically isomorphic to _p^*/q^ℤ (Tate curve) for some p-adic integer q. The behaviour is then as follows:

* If the reduction is split (E.tate[2] is a t_PADIC), we have an isomorphism φ: E(ℚ_p) ~ ℚ_p^*/q^ℤ and the function returns φ(P) ∈ ℚ_p.

* If the reduction is not split (E.tate[2] is a t_POLMOD), we only have an isomorphism φ: E(K) ~ K^*/q^ℤ over the unramified quadratic extension K/ℚ_p. In this case, the output φ(P) ∈ K is a t_POLMOD.

  ? E = ellinit([0,-1,1,0,0], O(11^5)); P = [0,0];
  ? [u2,u,q] = E.tate; type(u) \\ split multiplicative reduction
  %2 = "t_PADIC"
  ? ellmul(E, P, 5)  \\ P has order 5
  %3 = [0]
  ? z = ellpointtoz(E, [0,0])
  %4 = 3 + 11^2 + 2*11^3 + 3*11^4 + 6*11^5 + 10*11^6 + 8*11^7 + O(11^8)
  ? z^5
  %5 = 1 + O(11^9)
  ? E = ellinit(ellfromj(1/4), O(2^6)); x=1/2; y=ellordinate(E,x)[1];
  ? z = ellpointtoz(E,[x,y]); \\ t_POLMOD of t_POL with t_PADIC coeffs
  ? liftint(z) \\ lift all p-adics
  %8 = Mod(8*u + 7, u^2 + 437)

The library syntax is GEN zell(GEN E, GEN P, long prec).


ellpow(E, z, n)

Deprecated alias for ellmul.

The library syntax is GEN ellmul(GEN E, GEN z, GEN n).


ellratpoints(E, h, {flag = 0})

E being an integral model of elliptic curve , return a vector containing the affine rational points on the curve of naive height less than h. If flag = 1, stop as soon as a point is found; return either an empty vector or a vector containing a single point. See hyperellratpoints for how h can be specified.

  ? E=ellinit([-25,1]);
  ? ellratpoints(E,10)
  %2 = [[-5,1],[-5,-1],[-3,7],[-3,-7],[-1,5],[-1,-5],
        [0,1],[0,-1],[5,1],[5,-1],[7,13],[7,-13]]
  ? ellratpoints(E,10,1)
  %3 = [[-5,1]]

The library syntax is GEN ellratpoints(GEN E, GEN h, long flag).


ellrootno(E, {p})

E being an ell structure over ℚ as output by ellinit, this function computes the local root number of its L-series at the place p (at the infinite place if p = 0). If p is omitted, return the global root number and in this case the curve can also be defined over a number field.

Note that the global root number is the sign of the functional equation and conjecturally is the parity of the rank of the Mordell-Weil group. The equation for E needs not be minimal at p, but if the model is already minimal the function will run faster.

The library syntax is long ellrootno(GEN E, GEN p = NULL).


ellsea(E, {tors = 0})

Let E be an ell structure as output by ellinit, defined over a finite field 𝔽_q. This low-level function computes the order of the group E(𝔽_q) using the SEA algorithm; compared to the high-level function ellcard, which includes SEA among its choice of algorithms, the tors argument allows to speed up a search for curves having almost prime order. When tors is set to a non-zero value, the function returns 0 as soon as it detects that the order has a small prime factor not dividing tors; SEA considers modular polynomials of increasing prime degree ℓ and we return 0 as soon as we hit an ℓ (coprime to tors) dividing #E(𝔽_q):

  ? ellsea(ellinit([1,1], 2^56+3477), 1)
  %1 = 72057594135613381
  ? forprime(p=2^128,oo, q = ellcard(ellinit([1,1],p)); if(isprime(q),break))
  time = 6,571 ms.
  ? forprime(p=2^128,oo, q = ellsea(ellinit([1,1],p),1);if(isprime(q),break))
  time = 522 ms.

In particular, set tors to 1 if you want a curve with prime order, to 2 if you want to allow a cofactor which is a power of two (e.g. for Edwards's curves), etc. The early exit on bad curves yields a massive speedup compared to running the cardinal algorithm to completion.

The following function returns a curve of prime order over 𝔽_p.

  cryptocurve(p) =
  {
    while(1,
      my(E, N, j = Mod(random(p), p));
      E = ellinit(ellfromj(j));
      N = ellsea(E, 1); if (!N, continue);
      if (isprime(N), return(E));
      \\ try the quadratic twist for free
      if (isprime(2*p+2 - N), return(ellinit(elltwist(E))));
    );
  }
  ? p = randomprime([2^255, 2^256]);
  ? E = cryptocurve(p); \\ insist on prime order
  %2 = 47,447ms

The same example without early abort (using ellcard(E) instead of ellsea(E, 1)) runs for about 5 minutes before finding a suitable curve.

The availability of the seadata package will speed up the computation, and is strongly recommended. The generic function ellcard should be preferred when you only want to compute the cardinal of a given curve without caring about it having almost prime order:

* If the characteristic is too small (p ≤ 7) or the field cardinality is tiny (q ≤ 523) the generic algorithm ellcard is used instead and the tors argument is ignored. (The reason for this is that SEA is not implemented for p ≤ 7 and that if q ≤ 523 it is likely to run into an infinite loop.)

* If the field cardinality is smaller than about 250, the generic algorithm will be faster.

* Contrary to ellcard, ellsea does not store the computed cardinality in E.

The library syntax is GEN ellsea(GEN E, ulong tors).


ellsearch(N)

This function finds all curves in the elldata database satisfying the constraint defined by the argument N:

* if N is a character string, it selects a given curve, e.g. "11a1", or curves in the given isogeny class, e.g. "11a", or curves with given conductor, e.g. "11";

* if N is a vector of integers, it encodes the same constraints as the character string above, according to the ellconvertname correspondance, e.g. [11,0,1] for "11a1", [11,0] for "11a" and [11] for "11";

* if N is an integer, curves with conductor N are selected.

If N codes a full curve name, for instance "11a1" or [11,0,1], the output format is [N, [a_1,a_2,a_3,a_4,a_6], G] where [a_1,a_2,a_3,a_4,a_6] are the coefficients of the Weierstrass equation of the curve and G is a ℤ-basis of the free part of the Mordell-Weil group attached to the curve.

  ? ellsearch("11a3")
  %1 = ["11a3", [0, -1, 1, 0, 0], []]
  ? ellsearch([11,0,3])
  %2 = ["11a3", [0, -1, 1, 0, 0], []]

If N is not a full curve name, then the output is a vector of all matching curves in the above format:

  ? ellsearch("11a")
  %1 = [["11a1", [0, -1, 1, -10, -20], []],
        ["11a2", [0, -1, 1, -7820, -263580], []],
        ["11a3", [0, -1, 1, 0, 0], []]]
  ? ellsearch("11b")
  %2 = []

The library syntax is GEN ellsearch(GEN N). Also available is GEN ellsearchcurve(GEN N) that only accepts complete curve names (as t_STR).


ellsigma(L, {z = 'x}, {flag = 0})

Computes the value at z of the Weierstrass σ function attached to the lattice L as given by ellperiods(,1): including quasi-periods is useful, otherwise there are recomputed from scratch for each new z. σ(z, L) = z ∏ω ∈ L^* (1 - (z)/(ω))e(z)/(ω) + (z^2)/(2ω^2). It is also possible to directly input L = [ω_1,ω_2], or an elliptic curve E as given by ellinit (L = E.omega).

  ? w = ellperiods([1,I], 1);
  ? ellsigma(w, 1/2)
  %2 = 0.47494937998792065033250463632798296855
  ? E = ellinit([1,0]);
  ? ellsigma(E) \\ at 'x, implicitly at default seriesprecision
  %4 = x + 1/60*x^5 - 1/10080*x^9 - 23/259459200*x^13 + O(x^17)

If flag = 1, computes an arbitrary determination of log(σ(z)).

The library syntax is GEN ellsigma(GEN L, GEN z = NULL, long flag, long prec).


ellsub(E, z1, z2)

Difference of the points z1 and z2 on the elliptic curve corresponding to E.

The library syntax is GEN ellsub(GEN E, GEN z1, GEN z2).


elltamagawa(E)

The object E being an elliptic curve over a number field, returns the global Tamagawa number of the curve (including the factor at infinite places).

  ? e = ellinit([1, -1, 1, -3002, 63929]); \\ curve "90c6" from elldata
  ? elltamagawa(e)
  %2 = 288
  ? [elllocalred(e,p)[4] | p<-[2,3,5]]
  %3 = [6, 4, 6]
  ? vecprod(%)  \\ since e.disc > 0 the factor at infinity is 2
  %4 = 144

The library syntax is GEN elltamagawa(GEN E).


elltaniyama(E, {d = seriesprecision})

Computes the modular parametrization of the elliptic curve E/ℚ, where E is an ell structure as output by ellinit. This returns a two-component vector [u,v] of power series, given to d significant terms (seriesprecision by default), characterized by the following two properties. First the point (u,v) satisfies the equation of the elliptic curve. Second, let N be the conductor of E and Φ: X_0(N) → E be a modular parametrization; the pullback by Φ of the Néron differential du/(2v+a_1u+a_3) is equal to 2iπ f(z)dz, a holomorphic differential form. The variable used in the power series for u and v is x, which is implicitly understood to be equal to exp(2iπ z).

The algorithm assumes that E is a strong Weil curve and that the Manin constant is equal to 1: in fact, f(x) = ∑n > 0 ellan(E, n) x^n.

The library syntax is GEN elltaniyama(GEN E, long precdl).


elltatepairing(E, P, Q, m)

Computes the Tate pairing of the two points P and Q on the elliptic curve E. The point P must be of m-torsion.

The library syntax is GEN elltatepairing(GEN E, GEN P, GEN Q, GEN m).


elltors(E)

If E is an elliptic curve defined over a number field or a finite field, outputs the torsion subgroup of E as a 3-component vector [t,v1,v2], where t is the order of the torsion group, v1 gives the structure of the torsion group as a product of cyclic groups (sorted by decreasing order), and v2 gives generators for these cyclic groups. E must be an ell structure as output by ellinit.

  ?  E = ellinit([-1,0]);
  ?  elltors(E)
  %1 = [4, [2, 2], [[0, 0], [1, 0]]]

Here, the torsion subgroup is isomorphic to ℤ/2ℤ x ℤ/2ℤ, with generators [0,0] and [1,0].

The library syntax is GEN elltors(GEN E).


elltwist(E, {P})

Returns the coefficients [a_1,a_2,a_3,a_4,a_6] of the twist of the elliptic curve E by the quadratic extension of the coefficient ring defined by P (when P is a polynomial) or quadpoly(P) when P is an integer. If E is defined over a finite field, then P can be omitted, in which case a random model of the unique non-trivial twist is returned. If E is defined over a number field, the model should be replaced by a minimal model (if one exists).

Example: Twist by discriminant -3:

  ? elltwist(ellinit([0,a2,0,a4,a6]),-3)
  %1 = [0,-3*a2,0,9*a4,-27*a6]

Twist by the Artin-Shreier extension given by x^2+x+T in characteristic 2:

  ? lift(elltwist(ellinit([a1,a2,a3,a4,a6]*Mod(1,2)),x^2+x+T))
  %1 = [a1,a2+a1^2*T,a3,a4,a6+a3^2*T]

Twist of an elliptic curve defined over a finite field:

  ? E=ellinit([1,7]*Mod(1,19));lift(elltwist(E))
  %1 = [0,0,0,11,12]

The library syntax is GEN elltwist(GEN E, GEN P = NULL).


ellweilpairing(E, P, Q, m)

Computes the Weil pairing of the two points of m-torsion P and Q on the elliptic curve E.

The library syntax is GEN ellweilpairing(GEN E, GEN P, GEN Q, GEN m).


ellwp(w, {z = 'x}, {flag = 0})

Computes the value at z of the Weierstrass ℘ function attached to the lattice w as given by ellperiods. It is also possible to directly input w = [ω_1,ω_2], or an elliptic curve E as given by ellinit (w = E.omega).

  ? w = ellperiods([1,I]);
  ? ellwp(w, 1/2)
  %2 = 6.8751858180203728274900957798105571978
  ? E = ellinit([1,1]);
  ? ellwp(E, 1/2)
  %4 = 3.9413112427016474646048282462709151389

One can also compute the series expansion around z = 0:

  ? E = ellinit([1,0]);
  ? ellwp(E)              \\ 'x implicitly at default seriesprecision
  %5 = x^-2 - 1/5*x^2 + 1/75*x^6 - 2/4875*x^10 + O(x^14)
  ? ellwp(E, x + O(x^12)) \\ explicit precision
  %6 = x^-2 - 1/5*x^2 + 1/75*x^6 + O(x^9)

Optional flag means 0 (default): compute only ℘(z), 1: compute [℘(z),℘'(z)].

The library syntax is GEN ellwp0(GEN w, GEN z = NULL, long flag, long prec). For flag = 0, we also have GEN ellwp(GEN w, GEN z, long prec), and GEN ellwpseries(GEN E, long v, long precdl) for the power series in variable v.


ellxn(E, n, {v = 'x})

In standard notation, for any affine point P = (v,w) on the curve E, we have [n]P = (φ_n(P)ψ_n(P) : ω_n(P) : ψ_n(P)^3) for some polynomials φ_n,ω_n,ψ_n in ℤ[a_1,a_2,a_3,a_4,a_6][v,w]. This function returns [φ_n(P),ψ_n(P)^2], which give the numerator and denominator of the abscissa of [n]P and depend only on v.

  ? E = ellinit([17,42]);
  ? T = ellxn(E, 2, 'X)
  %2 = [X^4 - 34*X^2 - 336*X + 289, 4*X^3 + 68*X + 168]
  ? P = [114,1218]; ellmul(E,P,2)
  %3 = [200257/7056, 90637343/592704]
  ? [x,y] = subst(T,'X,P[1]) \\ substitute P[1] in ellxn(E,2)
  %4 = [168416137, 5934096] \\ numerator and denominator of 2*P
  ? x/y                     \\ check we find ellmul(e,P,2)[1]
  %5 = 200257/7056

The library syntax is GEN ellxn(GEN E, long n, long v = -1) where v is a variable number.


ellzeta(w, {z = 'x})

Computes the value at z of the Weierstrass ζ function attached to the lattice w as given by ellperiods(,1): including quasi-periods is useful, otherwise there are recomputed from scratch for each new z. ζ(z, L) = (1)/(z) + z^2∑ω ∈ L^* (1)/(ω^2(z-ω)). It is also possible to directly input w = [ω_1,ω_2], or an elliptic curve E as given by ellinit (w = E.omega). The quasi-periods of ζ, such that ζ(z + aω_1 + bω_2) = ζ(z) + aη_1 + bη_2 for integers a and b are obtained as η_i = 2ζ(ω_i/2). Or using directly elleta.

  ? w = ellperiods([1,I],1);
  ? ellzeta(w, 1/2)
  %2 = 1.5707963267948966192313216916397514421
  ? E = ellinit([1,0]);
  ? ellzeta(E, E.omega[1]/2)
  %4 = 0.84721308479397908660649912348219163647

One can also compute the series expansion around z = 0 (the quasi-periods are useless in this case):

  ? E = ellinit([0,1]);
  ? ellzeta(E) \\ at 'x, implicitly at default seriesprecision
  %4 = x^-1 + 1/35*x^5 - 1/7007*x^11 + O(x^15)
  ? ellzeta(E, x + O(x^20)) \\ explicit precision
  %5 = x^-1 + 1/35*x^5 - 1/7007*x^11 + 1/1440257*x^17 + O(x^18)

The library syntax is GEN ellzeta(GEN w, GEN z = NULL, long prec).


ellztopoint(E, z)

E being an ell as output by ellinit, computes the coordinates [x,y] on the curve E corresponding to the complex or p-adic parameter z. Hence this is the inverse function of ellpointtoz.

* If E is defined over a p-adic field and has multiplicative reduction, then z is understood as an element on the Tate curve Q_p^* / q^ℤ.

  ? E = ellinit([0,-1,1,0,0], O(11^5));
  ? [u2,u,q] = E.tate; type(u)
  %2 = "t_PADIC" \\ split multiplicative reduction
  ? z = ellpointtoz(E, [0,0])
  %3 = 3 + 11^2 + 2*11^3 + 3*11^4 + 6*11^5 + 10*11^6 + 8*11^7 + O(11^8)
  ? ellztopoint(E,z)
  %4 = [O(11^9), O(11^9)]
  
  ? E = ellinit(ellfromj(1/4), O(2^6)); x=1/2; y=ellordinate(E,x)[1];
  ? z = ellpointtoz(E,[x,y]); \\ non-split: t_POLMOD with t_PADIC coefficients
  ? P = ellztopoint(E, z);
  ? P[1] \\ y coordinate is analogous, more complicated
  %8 = Mod(O(2^4)*x + (2^-1 + O(2^5)), x^2 + (1 + 2^2 + 2^4 + 2^5 + O(2^7)))

* If E is defined over the complex numbers (for instance over ℚ), z is understood as a complex number in ℂ/Λ_E. If the short Weierstrass equation is y^2 = 4x^3 - g_2x - g_3, then [x,y] represents the Weierstrass ℘-function and its derivative. For a general Weierstrass equation we have x = ℘(z) - b_2/12, y = ℘'(z) - (a_1 x + a_3)/2. If z is in the lattice defining E over ℂ, the result is the point at infinity [0].

  ? E = ellinit([0,1]); P = [2,3];
  ? z = ellpointtoz(E, P)
  %2 = 3.5054552633136356529375476976257353387
  ? ellwp(E, z)
  %3 = 2.0000000000000000000000000000000000000
  ? ellztopoint(E, z) - P
  %4 = [2.548947057811923643 E-57, 7.646841173435770930 E-57]
  ? ellztopoint(E, 0)
  %5 = [0] \\ point at infinity

The library syntax is GEN pointell(GEN E, GEN z, long prec).


genus2red(PQ, {p})

Let PQ be a polynomial P, resp. a vector [P,Q] of polynomials, with rational coefficients. Determines the reduction at p > 2 of the (proper, smooth) genus 2 curve C/ℚ, defined by the hyperelliptic equation y^2 = P(x), resp. y^2 + Q(x)*y = P(x). (The special fiber X_p of the minimal regular model X of C over ℤ.)

If p is omitted, determines the reduction type for all (odd) prime divisors of the discriminant.

This function was rewritten from an implementation of Liu's algorithm by Cohen and Liu (1994), genus2reduction-0.3, see http://www.math.u-bordeaux.fr/~liu/G2R/.

CAVEAT. The function interface may change: for the time being, it returns [N,FaN, T, V] where N is either the local conductor at p or the global conductor, FaN is its factorization, y^2 = T defines a minimal model over ℤ[1/2] and V describes the reduction type at the various considered p. Unfortunately, the program is not complete for p = 2, and we may return the odd part of the conductor only: this is the case if the factorization includes the (impossible) term 2-1; if the factorization contains another power of 2, then this is the exact local conductor at 2 and N is the global conductor.

  ? default(debuglevel, 1);
  ? genus2red(x^6 + 3*x^3 + 63, 3)
  (potential) stable reduction: [1, []]
  reduction at p: [III{9}] page 184, [3, 3], f = 10
  %1 = [59049, Mat([3, 10]), x^6 + 3*x^3 + 63, [3, [1, []],
         ["[III{9}] page 184", [3, 3]]]]
  ? [N, FaN, T, V] = genus2red(x^3-x^2-1, x^2-x);  \\ X_1(13), global reduction
  p = 13
  (potential) stable reduction: [5, [Mod(0, 13), Mod(0, 13)]]
  reduction at p: [I{0}-II-0] page 159, [], f = 2
  ? N
  %3 = 169
  ? FaN
  %4 = Mat([13, 2])   \\ in particular, good reduction at 2 !
  ? T
  %5 = x^6 + 58*x^5 + 1401*x^4 + 18038*x^3 + 130546*x^2 + 503516*x + 808561
  ? V
  %6 = [[13, [5, [Mod(0, 13), Mod(0, 13)]], ["[I{0}-II-0] page 159", []]]]

We now first describe the format of the vector V = V_p in the case where p was specified (local reduction at p): it is a triple [p, stable, red]. The component stable = [type, vecj] contains information about the stable reduction after a field extension; depending on types, the stable reduction is

* 1: smooth (i.e. the curve has potentially good reduction). The Jacobian J(C) has potentially good reduction.

* 2: an elliptic curve E with an ordinary double point; vecj contains j mod p, the modular invariant of E. The (potential) semi-abelian reduction of J(C) is the extension of an elliptic curve (with modular invariant j mod p) by a torus.

* 3: a projective line with two ordinary double points. The Jacobian J(C) has potentially multiplicative reduction.

* 4: the union of two projective lines crossing transversally at three points. The Jacobian J(C) has potentially multiplicative reduction.

* 5: the union of two elliptic curves E_1 and E_2 intersecting transversally at one point; vecj contains their modular invariants j_1 and j_2, which may live in a quadratic extension of 𝔽_p and need not be distinct. The Jacobian J(C) has potentially good reduction, isomorphic to the product of the reductions of E_1 and E_2.

* 6: the union of an elliptic curve E and a projective line which has an ordinary double point, and these two components intersect transversally at one point; vecj contains j mod p, the modular invariant of E. The (potential) semi-abelian reduction of J(C) is the extension of an elliptic curve (with modular invariant j mod p) by a torus.

* 7: as in type 6, but the two components are both singular. The Jacobian J(C) has potentially multiplicative reduction.

The component red = [NUtype, neron] contains two data concerning the reduction at p without any ramified field extension.

The NUtype is a t_STR describing the reduction at p of C, following Namikawa-Ueno, The complete classification of fibers in pencils of curves of genus two, Manuscripta Math., vol. 9, (1973), pages 143-186. The reduction symbol is followed by the corresponding page number or page range in this article.

The second datum neron is the group of connected components (over an algebraic closure of 𝔽_p) of the Néron model of J(C), given as a finite abelian group (vector of elementary divisors).

If p = 2, the red component may be omitted altogether (and replaced by [], in the case where the program could not compute it. When p was not specified, V is the vector of all V_p, for all considered p.

Notes about Namikawa-Ueno types.

* A lower index is denoted between braces: for instance, [I{2}-II-5] means [I_2-II-5].

* If K and K' are Kodaira symbols for singular fibers of elliptic curves, then [K-K'-m] and [K'-K-m] are the same.

We define a total ordering on Kodaira symbol by fixing I < I* < II < II*,.... If the reduction type is the same, we order by the number of components, e.g. I_2 < I_4, etc. Then we normalize our output so that K ≤ K'.

* [K-K'--1] is [K-K'-α] in the notation of Namikawa-Ueno.

* The figure [2I_0-m] in Namikawa-Ueno, page 159, must be denoted by [2I_0-(m+1)].

The library syntax is GEN genus2red(GEN PQ, GEN p = NULL).


hyperellcharpoly(X)

X being a non-singular hyperelliptic curve defined over a finite field, return the characteristic polynomial of the Frobenius automorphism. X can be given either by a squarefree polynomial P such that X: y^2 = P(x) or by a vector [P,Q] such that X: y^2 + Q(x) y = P(x) and Q^2+4 P is squarefree.

The library syntax is GEN hyperellcharpoly(GEN X).


hyperellpadicfrobenius(Q, p, n)

Let X be the curve defined by y^2 = Q(x), where Q is a polynomial of degree d over ℚ and p ≥ d a prime such that X has good reduction at p return the matrix of the Frobenius endomorphism ϕ on the crystalline module D_p(X) = ℚ_p ⨂ H^1dR(X/ℚ) with respect to the basis of the given model (ω, x ω,...,xg-1 ω), where ω = dx/(2 y) is the invariant differential, where g is the genus of X (either d = 2 g+1 or d = 2 g+2). The characteristic polynomial of ϕ is the numerator of the zeta-function of the reduction of the curve X modulo p. The matrix is computed to absolute p-adic precision p^n.

The library syntax is GEN hyperellpadicfrobenius(GEN Q, ulong p, long n).


hyperellratpoints(X, h, {flag = 0})

X being a non-singular hyperelliptic curve given by an integral model, return a vector containing the affine rational points on the curve of naive height less than h. If flag = 1, stop as soon as a point is found; return either an empty vector or a vector containing a single point.

X is given either by a squarefree polynomial P such that X: y^2 = P(x) or by a vector [P,Q] such that X: y^2+Q(x) y = P(x) and Q^2+4 P is squarefree.

The parameter h can be

* an integer H: find the points [n/d,y] whose abscissas x = n/d have naive height ( = max(|n|, d)) less than H;

* a vector [N,D] with D ≤ N: find the points [n/d,y] with |n| ≤ N, d ≤ D.

* a vector [N,[D_1,D_2]] with D_1 < D_2 ≤ N find the points [n/d,y] with |n| ≤ N and D_1 ≤ d ≤ D_2.

The library syntax is GEN hyperellratpoints(GEN X, GEN h, long flag).