Re: general educational question on elliptic curve isogenies and moving points around

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

Using

K = [[1, 0, 0, -1473670506991667240419073314040, 
487857899055649012085793427889299493072289600], [1, 0, 0, 
-1350882210032036138970816514040, 
604255810745174408864225385407951158557249600], [1, 0, 0, 
-21613375759952551922130559714040, 
38675151136497269167819387070355078625642209600], [1, 0, 0, 
-76802094602026958682816514040, 
11216788650446842210221197792454992157249600], [1, 0, 0, 
3910584245041201096206610543960, 
3229366054013312573785301403224849373420037200], [1, 0, 0, 
-8822538010378633200216865972040, 
-9703116604031639452280171990115551185285698000], [1, 0, 0, 
4438136821871996939736684043660, 
-36328127880290265739543247394077097481682050860], [1, 0, 0, 
-139665092896820718696935098515740, 
-635300473525359445488968869790600609112629919540]]
which is an isogeny list of 8 curves, I cannot get ellisisom(E,F) to 
even work out all all

All results = 0 for using K[1] and trying the other 7 curves.

for example ellisisom(K[1],K[2]) = 0

What am I doing wrong?

On 11/25/23 15:29, Bill Allombert wrote:
> On Sat, Nov 25, 2023 at 01:32:55PM -0800, American Citizen wrote:
> > Hello:
> >
> > I am trying to carefully learn about elliptic curve isogenies, and how to
> > move points across two isogenous curves. (which have the same conductor, of
> > course)
> >
> > Suppose I have a non-minimal curve, say E with point P
> >
> > I have another curve F, and I want to find the image of P on F.
> How did you find F ? Usually you should have more information of the isogeny,
> such as the degree, that will help you.
>
> ellisogeny provides "isogeny from kernel", but GP is lacking the function
> "isogeny from image" because we are not quite sure what should be the
> specification. (However this is implemented in PARI for ellisomat).
>
> Also we do not have ellisisomorphic but you can use this:
>
> ellisisom(E,F)=
> {
>    my(u,r,s,t);
>    if (E.j!=F.j, return(0));
>    if (E.j==0,
>         if(!ispower(F.c6/E.c6,6,&u),return(0))
>       ,E.j==1728,
>         if(!ispower(F.c4/E.c4,4,&u),return(0))
>       , if(!ispower(F.c4*E.c6/(F.c6*E.c4),2,&u),return(0)));
>    s = (u*F.a1-E.a1)/2;
>    r = (u^2*F.a2 + s*E.a1 - E.a2 + s^2)/3;
>    t = (u^3*F.a3 - r*E.a1 - E.a3)/2;
>    [u,r,s,t]
> }
>
> So you can call ellisomat of E, find the curve which is isomorphic to F and compute
> the isomorphism.
>
> Cheers,
> Bill.