tossing out an idea to use heights of algebraic points on elliptic curves to help sort isogenies for point searches

[American Citizen <website.reader3@gmail.com>]

I am tossing out an idea that I have had for several years now, to get 
your reaction to it.

In searching for rational points on some of the curves with very large 
conductors, such as the Z2xZ8 or Z2xZ4 torsion groups, I have used the 
isogenous curves to an advantage, selecting certain ones because the 
point heights for the Mordell-Weil basis are lower.

Let's look at one of my typical curves for face cuboids:

py:{4,1}+ (15,8) Elliptic Curve: [0,-31679,0,-1493049600,0]
 Rank:1 Tate-Sha:1 Regulator:4.0697304843631873930432752505566566024
 W1:[78408,13046616]
 Generator:26/7,15/8,190/99 Height:4.0697304843631873930432752505566566024
 Pentacycle:[26,7],[15,8],[190,99],[627,161],[299,49]
 Torsion: Z4xZ2 Pts: [ [0], [0, 0], [-11760, 3398640], [57600, 0], 
[-11760, -3398640], [126960, 36691440], [-25921, 0], [126960, -36691440] ]

From the curve E = [0, -31679, 0, -149309600,0] we find the minimal 
model curve and from that we find the 8-isgenous curves:

 Minimal model: M = [1, 0, 0, -114223080, -283150929600]

               8 isogenous curves:                L-series Reg       MW 
Point                      SHA

 I1 = [1, 0, 0,   -114223080,     -283150929600]   4.0697304844 [16962, 
1622346]                   1
 I2 = [1, 0, 0,    -49423080,      130545230400]   2.0348652422 [3480, 
24720]                      1
 I3 = [1, 0, 0,    -49095400,      132402324032]   4.0697304844 [4724, 
74468]                      1
 I4 = [1, 0, 0,     10134040,      425388708672]   4.0697304844 [238, 
653956]                      1
 I5 = [1, 0, 0,    352354920,    -2007529902000]   8.1394609687 [7350, 
985950]                     1
 I6 = [1, 0, 0,  -1617601080,   -25035066997200]  32.5578438749 [74762, 
16452092]                  4
 I7 = [1, 0, 0,  -1409434380,   -31714678433460] 260.4627509993 
[12562748/169, 36303784154/2197]  16
 I8 = [1, 0, 0, -25879815780, -1602472070375340] 260.4627509993 
[12058568/49, 28516665638/343]    16

I1 = M, of course.

I have found from past experience that the last 2 or 4 isogenous curves 
usually have SHA > 1, usually 2^n where n={2,4,etc}

Using the isogeny map from:

 isog(F,E)->my(S=ellisomat(E)[1]);my([f,fd,urst]=findisom(S,F));[P->ellchangepoint(ellisogenyapply(f,P),urst),P->ellisogenyapply(fd,ellchangepointinv(P,urst))]

We move point p = [16962, 1622346] on I1 to the other 7 isogenous 
curves. These points are not necessarily reduced.

  point p2 on I2 = [1998996/361, 1101130188/6859] height:8.139+
  point p3 on I3 = [1239866848/303601, 368574683576/167284151] 
height:16.2789+
  point p4 on I4 = [-93455084/225625, 69570063653164/107171875] 
height:16.2789+
  point p5 on I5 = [13287418/1089, 72621008602/35937] height:8.139+
  point p6 on I6 = [74762, -16526854] height:8.139+
  point p7 on I7 = [12562748/169, -36467099878/2197] height:8.139+
  point p8 on I8 = [12058568/49, 28516665638/343] height:16.2789+

This was done to show that the isogeny maps found do actually work.

Now lets take a non-rational point on M (or I1) say P1 = [-2641, 
(2641+sqrt(373230717)/2] = approx = [-2641, 
10980.0900145917166146328056490104100484287144219040808189].

This point is an algebraic point. Lets move it to the other 7 isogenies 
using the maps. After some careful work and adjustment of GP-Pari 
precision to successfully identifie the surds, we find that:

  P1 = [-2641, (2641+sqrt(373230717)/2]

  P2 =  [118195921/25917, 
-4964.59478568773289487300570684264312039574845396833456229]
                                   Or rather
  P2 =  [118195921/25917, 
-118195921/51834-186657119/1343381778*sqrt(373230717)]

  P3 = [137656039630182673/34005844178493, 
-2051.180754833907121481575845351035876153760852543913484078793783294978327248669742402457018713025089265830324249400885664886831917994514274772346445584977331412146788331217396915986870331920505804389535685346564864345995104211708279698629930578244267]
  P3 = [137656039630182673/34005844178493, 
-137656039630182673/68011688356986-89812271142122611231/63848794318082227206126*sqrt(373230717)]

  P4 = [-4862290343371247/688184701053, 
-29611.37634970070193846955359010625982113907516636543169216572480747151522903903035820881639158927877572287315269865633148465736792035181575735105149000355285996515546179661024480227553331517529781235835740488732840239070111670296357147325268390204820]
  P4 = [-4862290343371247/688184701053, 
4862290343371247/1376369402106-315353037098830011391/183814547938446333906*sqrt(373230717)]

  P5 = [93312959, 
901343791145.7248055161456534536014689255645919373468659667810330184948020327188402723657591901124117530141924558229632861159081575198774740510424666001074136802740735641524136420732288693604262411700208952370627933629378167015878103482808113860990604]
  P5 = [93312959, -93312959/2+93315601/2*sqrt(373230717)]

  P6 = [-338708641/14401, 
7414.861988651332552454406641825858904042992427037720399227171764668061175404296935414391078760131889971894300490220819127924158449042238767877955594946139255302700687262160015127607595147566635721846943664528830768300742147171111134170025790227486365]
  P6 = [-338708641/14401, 
338708641/28802-93286799/414777602*sqrt(373230717)]

  P7 = [5708617722978179/124391820121, 
-530121.8617747221260794522742751126742073602149708207408368818334312238673334853377643882292937838993768464954176606516369136093186207647438858532375531203989123188891273214773600303432517858198065518790749868243306061128996301933147151329511415141447]
  P7 = [5708617722978179/124391820121, 
-5708617722978179/248783640242-276429153265304304121/10529652883984498438*sqrt(373230717)]

  P8 = [-1347588258112645861/14508879057481, 
46402.89860757901292009837858224173943793369226054229297218075389646319273833482488878474538457406797310530826890689531542631111474398705986584838008684341970533508069256924933453625004313390786977725205394304913627471107036231273320896628252198402314]
  P8 = [-1347588258112645861/14508879057481, 
1347588258112645861/29017758114962-25557411883231812119/13264079361369254333642*sqrt(373230717)]

All these points, P1..P8 are algebraic.

Examining these points, the simplest algebraic expressions are P1 and 
P5. All 8 points P1..P8 have a rational or integer x-coordinate, 
interestingly enough.

I would like to find a height function of these points such that the 
possible regulator size of the curve might be found this way. I 
previously started work on a C++ program to do heights of points who are 
in the real decimal field (doubles, etc). I was able to get values which 
seemed to make sense (to me).

I have elliptic curves, where the conductor is so huge, there is simply 
no way to calculate the L-series values effectively using known techniques.

My idea is to use the heights of algebraic points to help point the way 
to the optimal isogenous curve to search for rational points.

Is this a valid idea, to pull algebraic points across isogenies, and use 
a height function to help indicate which isogenous curve to search?

- Randall