Re: getting strange ellheight() error for number field point and curve

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

Bill

Thank you for your reply.

I will follow your suggestion and rescript my gp-pari command.

Randall

On 12/5/23 14:33, Bill Allombert wrote:
> On Mon, Dec 04, 2023 at 05:25:32PM -0800, American Citizen wrote:
> > Hi:
> >
> > minimal m[1..5] = [1, 0, 0, -114223080, -283150929600]
> > pt=[29099, 4571616.001890359167852374954252793155119185143350555006071823347634514965562343989206397255762390603970823616017868766947483185076860275462218345407187946353537171824424767170799129381839037]
> > y=[-29099/2, 1/2, 84131656042917]
> > which is y = -29099/2 + 1/2 * sqrt(84131656042917)
> > p=[29099, Mod(1/2*a - 29099/2, a^2 - 84131656042917)]
> > point is on curve m [1, 0, 0, -114223080, -283150929600] --> 1
> > F[1] = [0, 0, 0, -5482707841/48, -244634179112639/864]
> > P[1] = [349189/12, Mod(1/2*a, a^2 - 84131656042917)]
> > is on curve: 1
> >
> > this works out good for all 8 curves and the point on the curve.
> >
> > NF = nfinit(a^2 - 84131656042917)
> > FN=ellinit(F[1],NF);
> > Then we try to find the height as ellheight(FN,pt)
> >    ***   at top-level: find_iso_hts(e)
> >    ***                 ^---------------
> >    ***   in function find_iso_hts: ... ellheight(FN,pt)");H[i]=
> >    ***   ellheight(FN,P[i]);pr
> >    ***   ^---------------------
> >    *** ellheight: impossible inverse in Fl_inv: Mod(0, 2).
> Alas, this is a bug in PARI
> (ellheight does not handle non-minimal models over a number field correctly).
>
> Until we fix it, I suggest you do
>
> NF = nfinit(a^2 - 84131656042917);
> F1 = ellinit([0, 0, 0, -5482707841/48, -244634179112639/864]);
> P1 = [349189/12, Mod(1/2*a, a^2 - 84131656042917)];
> MF1 = ellminimalmodel(F1,&urst);
> MFN = ellinit(MF1,NF);
> MP1 = ellchangepoint(P1,urst);
> ellisoncurve(MFN,MP1)
> ellheight(MFN,MP1)
>
> %8 = 20.912087636398176541299794149336289140
>
> Cheers,
> Bill.