compute ℓ-division fields for elliptic curves over finite fields

[yyyyx4]

EllipticCurve_number_field has a .division_field() method to compute the smallest field of definition of E[ℓ] for primes ℓ.

This patch generalizes the .division_field() method to finite fields, where the same algorithm works with minor adaptions.

The only nontrivial changes are in this commit: sagemath/sagetrac-mirror@731011c

Depends on #33937

CC: @defeo @JohnCremona @categorie

Component: elliptic curves

Author: Lorenz Panny

Branch/Commit: 3be7fe8

Reviewer: Travis Scrimshaw, John Cremona

Issue created by migration from https://trac.sagemath.org/ticket/33939

[sagetrac-git]

Changed commit from 308d5e1 to 6b7dbcd

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

3b6be5f	use tuple syntax for finite-field constructor
4789d59	move .division_field() up to EllipticCurve_field
731011c	support finite fields in .division_field()
6b7dbcd	rename p->l to avoid confusion with characteristic

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

e740b22

remove unused import

[sagetrac-git]

Changed commit from 6b7dbcd to e740b22

[tscrim]

comment:4

LGTM modulo documentation changes:

-        - ``\ell`` -- a prime number (an element of `\ZZ`).
-
-        - ``names`` -- (default: ``t``) a variable name for the division field.
-
+        - ``\ell`` -- a prime number (an element of `\ZZ`)
+        - ``names`` -- (default: ``t``) a variable name for the division field
         - ``map`` -- (default: ``False``) also return an embedding of
-          the :meth:`base_field` into the resulting field.
-
         - ``kwds`` -- additional keywords passed to
-          :func:`~sage.rings.polynomial.polynomial_element.Polynomial.splitting_field`.
+          :func:`~sage.rings.polynomial.polynomial_element.Polynomial.splitting_field`

[tscrim]

Reviewer: Travis Scrimshaw

[sagetrac-git]

Changed commit from e740b22 to 3be7fe8

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

f13b6ef	Merge tag '9.7.beta2' into public/generalize_elliptic_curve_division_field_to_finite_fields
247381b	doc tweaks
3be7fe8	move default value to argument list, where it belongs

[yyyyx4]

comment:7

Thanks. Done, plus a few other small tweaks.

[JohnCremona]

Changed reviewer from Travis Scrimshaw to Travis Scrimshaw, John Cremona

[JohnCremona]

comment:8

Looks good to me.

I was wondering whether (over any base field) it would be worthwhile first going up to the ell'th cyclotomic extension of the base field since the ell-splitting field will contain that. But that would take some experimentation, and should not be on this ticket.

[vbraun]

Changed branch from public/generalize_elliptic_curve_division_field_to_finite_fields to 3be7fe8