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

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

I was able to write a script which successfully found the isogeny and 
dual isogeny on the mimimal models of isomorphisms and mapped the point 
back successfully through the forward and reverse mapping.

I guess my only question is how to us the apply command to specifically 
apply  x.y value from a known point to the isogeny or its dual then 
creating the point [f/h^2, g/h^3] from that apply possibly all on one line.

Randall


On 11/25/23 13:32, 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.
>
> My general gp script go something like this.. trying to build this 
> script up carefully.
>
> 1. obtain conductors of E and F, using the ellglobalred()[1] command 
> and see if they match, if not exit
>
> if the conductors match, let u = ellglobalred(E)[2] and v = 
> ellglobalred(F)[2] which is the transformation matrices.
>
> 2. obtain M the minimal curve of E and using ellchangepoint(P,v) where 
> v=ellglobalred(E)[2] previously obtained from step 1 find point Mp on 
> M. (double check by using ellheight check
>
> 3. obtain N, the minimal curve of F
>
> This is where things get tricky.
>
> 4. Set R = ellisomat(M)[1] and set S = ellisomat(N)[1]
>
> 5. Carefully step over R[i] then carefully step over S[j], such that 
> R[i] E a4,a6 component matches S[j] Ea4,a6]
>
> 6. Using the forward f isogeny, from R[i], move Mp to the point on the 
> E a4,a6 curve. Watch out for f/g^2,f/h^3 transform
>
> 7. Using the dual isogeny g from S[j], move the point on E a4,a6 back 
> to N.
>
> 8. Using the ellchangepointinv(pts on N,v) to move the points from N 
> back onto K.
>
> Is this the correct way to do the isogeny?
>
> I do not know a direct way of phi(E) --> K where phi is the isogeny. I 
> thought of moving point P to the minimal curve M, then moving the 
> point Mp to minimal curve N using the isogeny, then using the dual 
> isogeny to push Np, back up to K.
>
> But this requires that I have to have identical a4,a6 terms for the 
> [0,0,0,a4,a6] curves, so I have to loop and look for a match if I 
> understand what is happening here.
>
> Do I have this whole process correct?
>
> Randall