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

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

Bill

Thanks for that sufficient reply. I was not aware that I didn't have to 
move to the minimal model of the curve.

However past experience has shown me that normal Gaussian reduction (via 
the ellheightmatrix) doesn't quite work right for the non-minimal curves 
Mordell-Weil basis set, sometimes Minkowski reduction is necessary to 
find the smallest MW basis (how we define "smallest" is a story in itself)

So I guess I got into a habit of using the minimal form for the curve.

Randall

On 11/27/23 09:46, Bill Allombert wrote:
> On Sun, Nov 26, 2023 at 05:07:18PM -0800, American Citizen wrote:
> > Bill
> >
> > What I am expecting as a function operator is something like
> >
> > map_iso(F,E,pt) = [operate_on_x, operate_on_y] where x=p[1] and y=p[2] using
> > the f,g,h polynomials from the ellisomat results. derived from the minimal
> > model maps of the elliptic curves F and E.
> >
> > I would like to see the operator function f given like a GP-Pari function
> > say q = f(F,E,p) which outputs point q on F.
> >
> > The way I worked this was take E (given elliptic curve), find M (minimal
> > model of E), move points to M, then find N (minimal model of F) and
> > carefully compare the ellisomat results for both elliptic curves M and N, to
> > find a match on the E(a4,a6) curves. Once this is found,  I move points on E
> > to M, then move points from M to N, then move the points from N to F. As you
> > all know, this has to be carefully done.
> There is no need to take minimal models.
>
> You can do this:
>
> \\ find the curve in S isomorphic to F
> findisom(S,F)=
> {
>    foreach(S,s,
>      my(urst=ellisisom(ellinit(s[1]),F));
>      if(urst,return([s[2],s[3],urst])));
> }
>
> \\ return [f,fd], f isogeny from E to F, and fd its dual.
> isog(E,F)=
> {
>    my(S=ellisomat(E)[1]);
>    my([f,fd,urst]=findisom(S,F));
>    [P->ellchangepoint(ellisogenyapply(f,P),urst),
>     P->ellisogenyapply(fd,ellchangepointinv(P,urst))];
> }
>
> For example:
>
> ? E=ellinit(K[1]);
> ? F=ellinit(K[2]);
> ? P=elltors(E).gen[1]
> %3 = [-221813002148660,28351785859134866849500]
> ? [f,fd]=isog(E,F);
> ? Q=f(P)
> %5 = [833820316449280,-7588256062930707186440]
> ? ellisoncurve(F,Q)
> %6 = 1
> ? ellorder(F,Q)
> %7 = 4
> ? R=fd(Q)
> %8 = [990529152804880,-495264576402440]
> ? R==ellmul(E,P,2)
> %9 = 1
>
> Cheers,
> Bill.