This PR removes the call to _points_fast_sqrt() for computing the set of all points as it (seems) to always be slower than using _points_via_group_structure().

While performing this change, I also attempted to clean up the code and simplify / clarify certain docstrings.

Justification

When a curve is defined over a prime order finite field with order larger than 50, the set of all points is computed using the group structure of the curve.

However, when a curve is implemented over a non-prime field or a field with order less than 50, the set of all points is instead computed by brute-force enumeration by checking all x-coordinates.

For all cases I have tested, this enumeration is slower.

Small characteristic, prime field

When p < 50:

sage: E = EllipticCurve(GF(11), [1,1])
sage: %timeit E._points_via_group_structure()
165 µs ± 8.19 µs per loop (mean ± std. dev. of 7 runs, 10,000 loops each)
sage: %timeit E._points_fast_sqrt()
366 µs ± 34.2 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)
sage: sorted(E._points_via_group_structure()) == sorted(E._points_fast_sqrt())
True

When p > 50

sage: E = EllipticCurve(GF(163), [1,1])
sage: %timeit E._points_via_group_structure()
2.49 ms ± 40.6 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
sage: %timeit E._points_fast_sqrt()
4.1 ms ± 112 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
sage: sorted(E._points_via_group_structure()) == sorted(E._points_fast_sqrt())
True

Small characteristic, non-prime field

When the order is smaller than 50:

sage: E = EllipticCurve(GF(2^5), [1,0,1,0,1])
sage: %timeit E._points_via_group_structure()
277 µs ± 10.2 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)
sage: %timeit E._points_fast_sqrt()
2.96 ms ± 135 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
sage: sorted(E._points_via_group_structure()) == sorted(E._points_fast_sqrt())
True

When the order is larger than 50:

sage: E = EllipticCurve(GF(11^3), [1,1])
sage: %timeit E._points_via_group_structure()
15.6 ms ± 1.17 ms per loop (mean ± std. dev. of 7 runs, 100 loops each)
sage: %timeit E._points_fast_sqrt()
71.4 ms ± 2.94 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)
sage: sorted(E._points_via_group_structure()) == sorted(E._points_fast_sqrt())
True

Further comments

• The commit which introduced _points_fast_sqrt() as a method implemented it for hyperelliptic curves over finite fields and then added HyperellipticCurve_finite_field as a parent to EllipticCurve_finite_field to inherit this function. With the removal of the slower _points_fast_sqrt(), this parent inheritance has been removed.

• ruff noticed that from sage.sets.set import Set was unused, so I removed it

📝 Checklist

• The title is concise and informative.
• 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.