Improve CM testing for elliptic curves over number fields

[JohnCremona]

📚 Description

This implements the Cremona-Sutherland algorithm (see :arxiv:2301.11169) for detecting Hilbert Class Polynomials and computing the corresponding CM discriminants. Using this, a new default algorithm for detecting CM j-invariants, and hence whether or not elliptic curves over number fields, is implemented. This is very much faster than the older methods for CM testing which were only fast enough to be reasonable over fields of degree less than 10; the new method works fine up to degree 100 and beyond (as some of the examples show).

Secondly, the caching of the computation of CM orders and discriminants and j-invariants for class numbers up to 100 has been made a lot more efficient through better caching. Before, if you asked for all discriminants with class number (say) 10, it computed all those with class number up to 10 and threw the rest away, so that even if you then asked for class number 9 it had to start from scratch; now that second call will be instantaneous. And If you then ask for class number 11, it will be faster than it was since much of the data needed will have been cached.

This closes #35147

📝 Checklist

• [x ] I have made sure that the title is self-explanatory and the description concisely explains the PR.
• [x ] I have linked an issue or discussion.
• [ x] I have created tests covering the changes.
• [ x] I have updated the documentation accordingly.

⌛ Dependencies

None. Based on 10.0.beta0

[JohnCremona]

I see doctest failures and am fixing them.

I suppose the point of having a "draft" PR is that all the automatic tests will run, then one makes it a real PR after fixing issues.

[JohnCremona]

I see doctest failures and am fixing them.

There are no failures when I run the tests locally so am letting the automatic build and test run again.

[JohnCremona]

I see doctest failures and am fixing them.

There are no failures when I run the tests locally so am letting the automatic build and test run again.

OK, there really were, so I have fixed them.

[codecov-commenter]

Codecov Report

Patch coverage: 84.91% and project coverage change: -0.04 ⚠️

Comparison is base (c00e6c2) 88.62% compared to head (d83aa3a) 88.58%.

❗ Current head d83aa3a differs from pull request most recent head f9c28ca. Consider uploading reports for the commit f9c28ca to get more accurate results

Additional details and impacted files

@@             Coverage Diff             @@
##           develop   #35166      +/-   ##
===========================================
- Coverage    88.62%   88.58%   -0.04%     
===========================================
  Files         2148     2140       -8     
  Lines       398855   397570    -1285     
===========================================
- Hits        353480   352192    -1288     
- Misses       45375    45378       +3     

Impacted Files	Coverage Δ
src/sage/quadratic_forms/special_values.py	96.29% <ø> (-0.05%)	⬇️
src/sage/schemes/elliptic_curves/cm.py	73.14% <81.57%> (-16.39%)	⬇️
...c/sage/schemes/elliptic_curves/ell_finite_field.py	90.92% <89.23%> (+1.93%)	⬆️
src/sage/quadratic_forms/binary_qf.py	93.28% <100.00%> (-0.02%)	⬇️
src/sage/rings/number_field/number_field.py	93.59% <100.00%> (-0.01%)	⬇️
src/sage/schemes/elliptic_curves/heegner.py	90.96% <100.00%> (-0.01%)	⬇️

... and 152 files with indirect coverage changes

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[roed314]

Generally looks good!

[JohnCremona]

I have now finished my edits in response to the referee

[JohnCremona]

I just fixed a trivial merge conflict which arose because this had been sitting idle for 2 weeks since I addressed all the revier's concerns. Please @roed314 or someone else update the review so that this can be merged!

[github-actions]

Documentation preview for this PR is ready! 🎉
Built with commit: f9c28ca

[JohnCremona]

This has now been sitting idle for 4 weeks since I addressed all the revier's concerns. Please @roed314 or someone else update the review so that this can be merged!

[roed314]

Thanks for the reminder! This looks good to me.

[JohnCremona]

Thanks, @roed314 !