general educational question on elliptic curve isogenies and moving points around

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

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