Compute composite degree (separable) isogenies of EllipticCurves

[grhkm21]

• Implement .isogenies_degree for EllipticCurves

📝 Checklist

• The title is concise, informative, and self-explanatory.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation accordingly.

⌛ Dependencies

[yyyyx4]

Other than the few small remarks, looks good to me!

[grhkm21]

Thanks for the review :)

[yyyyx4]

In addition to the smaller code remarks above below, I'm also wondering if this should be an iterator: The list can grow very quickly, so doing it as a depth-first iterator might be a good idea?

[grhkm21]

I swear I wrote some code similar to the previous two commits, but somehow didn't commit it, and it's not in my git stash either. Anyways, please kindly review it once again :) it's very short now.

[github-actions]

Documentation preview for this PR (built with commit f5d5834; changes) is ready! 🎉

[yyyyx4]

There are duplicates in the list as soon as any prime appears with multiplicity 3 or more in the degree:

sage: E = EllipticCurve(GF(11^2), [1,0])
sage: len(list(E.isogenies_degree(2^3)))
27
sage: len(set(E.isogenies_degree(2^3)))
15

To fix it, we could restrict to cyclic isogenies:

sage: P,Q = E.torsion_basis(2)
sage: len([phi for phi in E.isogenies_degree(2^3) if phi(P) or phi(Q)])
12
sage: len({phi for phi in E.isogenies_degree(2^3) if phi(P) or phi(Q)})
12

Returning only cyclic isogenies is arguably "better" anyway, since non-cyclic isogenies could be considered somewhat degenerate among the set of all isogenies of given degree. However, adding this check will make the code quite a bit more complicated. I'm unsure how to proceed.

[grhkm21]

Oh right, sorry for missing that. We can have a check to ensure we are not taking the dual isogeny of the previous taken one, since the prime degrees considered are sorted.

[hellman]

I think it would be useful to have an phi1.is_dual_of(phi2) method (up to isomorphism check). I also need something like for mitm.

But if we add it to the EllipticCurveHom class, it should be generic (e.g. factor the degree?), which could introduce an overhead.