Re: Some comments on Isogenous Curves, algebraic points on those curves and SHA

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Thu, Dec 14, 2023 at 04:56:51PM -0800, American Citizen wrote:
> To all (well those who like elliptic curves):
> 
> I want to make some comments about isogenous curves, the recent work I did
> with algebraic points on those curves, and the determination of SHA, which
> has caused me difficulties the past 3 decades when working with elliptic
> curves, and yes, I have worked with millions of curves, literally.
> 
> This is a rather lengthy commentary, so you might wish to just skip over it,
> if your interest is not elliptic curves.
> 
> Ever since Martin Gardner of Mathematical Games section of Scientific
> American got me hooked on the rational cuboid problem, I have been looking
> at the elliptic curves related to the 3 types of cuboids. It was John Leech
> who pointed out the elliptic curves involved.
> 
> John Leech and I spent useful correspondence up until his sudden decease on
> Sept 28, 1992.  I expanded upon a key table which he was working upon, which
> I call "pyfm" for Pythagorean Face pentacycles and associated elliptic
> curves. See https://www.ams.org/journals/mcom/1986-46-174/S0025-5718-1986-0829644-0/S0025-5718-1986-0829644-0.pdf
> for this particular discussion, particularly page 753.
> 
> Let me start with one such elliptic curve:
> 
> Starting information from a certain pentacycle and its associated elliptic
> curve(s)
> 
> py:{9,2}+ (77,36) [0,-9271247,0,-659737307363904,0]
>  Rank:1 Tate-Sha:1 Regulator:3.3147856512874987450598709350772059668
>  W:[-119064,8855385000]
>  Generator:96/79,77/36,147/11 Height:3.3147856512874987450598709350772059668
>  Pentacycle:[96,79],[77,36],[147,11],[113,7],[316,113]
> 
> Curve:
>       e = [0,-9271247,0,-659737307363904,0]
> 
> Point on the curve (Mordell-Weil basis for the rank 1 curve)
> 
>       p = [-119064,8855385000]

Let us start here.

E=ellinit([0,-9271247,0,-659737307363904,0]);
P = [-119064,8855385000];
[S,M]=ellisomat(E);
C=apply(e->ellinit(e[1]),S);
L=lfun(E,1,1);
Q=vector(#S,i,ellisogenyapply(S[i][2],P));
F=vector(#S,i,ellsaturation(C[i],[Q[i]],2)[1]);
F==Q
round(vector(#S,i,L/ellbsd(C[i])/ellheight(C[i],F[i])))
[1,1,1,4,4,4,64,64]

The conclusion is that, assuming P is indeed a generator of E, the
Sha of the twists are 1,1,1,4,4,4,64,64.

Practically speaking you should search points on the curve which has the largest
ellbsd. Note that this does not require to compute L.

Cheers,
Bill.