Isogeny small degree improvement

[user202729]

Allow the following to be called without error.

sage: EllipticCurve(GF(31).algebraic_closure(), [0, 6, 0, 1, 0]).isogenies_prime_degree(5)

Inspired from, but independent from, https://github.com/sagemath/sage/issues/38373 . Whatever consensus on that issue might be.

---

Would be good if someone can sanity check that the output is actually correct. (although it looks correct to me.)

Strangely enough there is

        if isinstance(F, sage.rings.abc.AlgebraicField):
            raise NotImplementedError("This code could be implemented for QQbar, but has not been yet.")

even though when I comment that part out, the code seems to work well.

:memo: Checklist

• [x] The title is concise and informative.
• [x] The description explains in detail what this PR is about.
• [x] I have linked a relevant issue or discussion.
• [x] I have created tests covering the changes.
• [x] I have updated the documentation and checked the documentation preview.

:hourglass: Dependencies

https://github.com/sagemath/sage/pull/38994 : both changes are needed to make it work.

[github-actions[bot]]

Documentation preview for this PR (built with commit 53847e44b974dcfbd33bb56a1b2645137b92c932; changes) is ready! :tada: This preview will update shortly after each push to this PR.

[JohnCremona]

About correctness: looks reasonable to me. Over GF(31) there are only 2 5-isogenies, the others are defined over GF(31^2) and the output from that looks just like this output. Perhaps it would be safer to have the doctest just report that there are six 5-isogenies, to guard against future versions making different choices? My familiarity with the supersingular case is less than perfect! I note that of the 6 isogenous curves there are only 3 isomorphism classes, E itself (@3) another @2 and another unique.

I think it is great that this works as it should making only a trivial change to the code in isogenies_small_degree which I wrote years ago.

Can you explain the need for the new copy methods?

[user202729]

The copy is from https://github.com/sagemath/sage/pull/38994 . Basically it's needed to make it work at all because of some implementation detail.

Specifically, some part of the code requires making a new isogeny with a different pre-composed isomorphism, so it calls the internal method _set_pre_isomorphism (which mutates the isogeny (!)), but it wants a modified copy so it does deepcopy on the isogeny. Without special handling, deepcopy on the isogeny copies all the members, including the base field.

I think it's desirable for arbitrary objects to be copyable anyway, so I can't see any harm. (Or at least if it's desirable to make algebraic closure of finite field not copyable then at least raise a meaningful error, but at the moment loads(dumps(GF(p).algebraic_closure())) makes a copy anyway)

It might separately be a good idea to refactor the code to avoid mutable state in isogeny objects (easy to introduce accidental bugs), but currently it's just internal implementation detail anyway.