Re: 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>]

More things for me to learn, particularly number fields.

Well I am glad that my basic math instincts have been correct.

On 11/30/23 01:59, Bill Allombert wrote:
> On Wed, Nov 29, 2023 at 06:50:02PM -0800, American Citizen wrote:
> > 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]
> >
> >   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))]
> Note that you could just have used either ellisomat directly or use ellisogeny
> to compute the isogenies associated to the torsion points. In both cases, you
> would have obtained directly the curves and the isogenies, without the need to
> try to match curves afterward.
>
> > 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:
> You could just use elliptic curves over number fields, this is fully supported,
> and use ellisogenyapply.
>
> M = ellinit([1, 0, 0, -114223080, -283150929600]);
> P1 = [-2641, (2641+Mod(a,a^2-373230717))/2];
> L=ellisomat(M)[1];
> F=ellinit(L[2][1]);
> iso = L[2][2];
> Q=ellisogenyapply(iso,P1)
> %10 = [157597441/8639,Mod(-746628476/671690889*a,a^2-373230717)]
> ellisoncurve(F,Q)
> %11 = 1
>
> > 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).
> PARI supports elliptic curves over number fields, so you can use ellheight
> over number fields:
>
> E=ellinit([0, -31679, 0, -149309600,0]);
> M=ellinit([1, 0, 0, -114223080, -283150929600]);
> P1 = [-2641, (2641+Mod(a,a^2-373230717))/2]
> ellisoncurve(M,P1)
> K=nfinit(a^2-373230717);
> MK=ellinit(M,K);
> ellheight(MK,P1)
> %21 = 18.351564077458714575387532095240588314
>
> Cheers,
> Bill.