gh-37967: Modify tests in `ell_point.py` to be more generic for Conway database update
As part of the Conway polynomials database update, two tests in
`schemes.elliptic_curves.ell_point` caused issues due to their
dependency on the precise implementation of the finite field
`GF(65537^2)`:
- The first test (of `.weil_pairing`) checks whether the Sage
implementation still works for an example of fixed `7282`-torsion points
- The second test (of `.tate_pairing`) checks whether the new
implementation, which uses PARI up to reduction of the pairing,
coincides with the old Sage implementation (still used e.g. in
SageMathCell, but not available in the current code anymore)
To help with the database update, we adapt these tests to not be
dependent on the exact minimal polynomial used for finite field
generation:
- We turn both tests into examples (with fixed moduli) for their
respective methods
- We modify the test of `.weil_pairing` to check for two randomly chosen
`7282`-torsion points whether `algorithm="sage"` and `algorithm="pari"`
yield the same result
Please let me know if you think other polynomial-independent tests are
better suited as replacements.
See sagemath/conway-polynomials#5 (especially [comment
10](https://github.com/sagemath/conway-
polynomials/pull/5#issuecomment-2101462886) and Update 2 in [comment
11](https://github.com/sagemath/conway-
polynomials/pull/5#issuecomment-2101498241)) for more details.
CC: @yyyyx4 @GiacomoPope @JohnCremona
URL: https://github.com/sagemath/sage/pull/37967
Reported by: Sebastian A. Spindler
Reviewer(s):
