Expand completeness checker for elliptic curves and number fields

[rvisser7]

This PR extends the completeness checker (lmfdb/utils/completeness.py) with a few more cases for elliptic curves, genus 2 curves, and number fields.

As always, any comments/feedback much appreciated! 🙂

• Elliptic curves: Since the LMFDB has all elliptic curves over Q with 7-smooth conductor, we can thus guarantee completeness when searching for elliptic curves over Q whose bad primes are contained in {2,3,5,7}. This hence addresses Drew's comment in #6773.

  https://beta.lmfdb.org/EllipticCurve/Q/?bad_quantifier=subset&bad_primes=2%2C3%2C5%2C7
  http://localhost:37777/EllipticCurve/Q/?bad_quantifier=subset&bad_primes=2%2C3%2C5%2C7

• Further to the above, we can also guarantee completeness when searching for elliptic curves over Q whose bad primes are contained in some fixed set $S$, when the maximum possible conductor $M = \prod_{p \in S} p^{e_p}$, with $e_2 = 8$, $e_3 = 5$, and $e_p = 2$ for $p > 3$, is at most 500 000. E.g. we can guarantee completeness for elliptic curves E/Q good away from $S =$ {23, 29} or $S =$ {701}.

  https://beta.lmfdb.org/EllipticCurve/Q/?bad_quantifier=subset&bad_primes=23%2C29
  http://localhost:37777/EllipticCurve/Q/?bad_quantifier=subset&bad_primes=23%2C29

• Genus 2 curves: We can (rather trivially) guarantee completeness for genus 2 curves over Q with conductor less than 121, since no such curves exist! Though (as far as I know) this requires assuming paramodularity.

  https://beta.lmfdb.org/Genus2Curve/Q/?cond=1-120
  http://localhost:37777/Genus2Curve/Q/?cond=1-120

• Regulator bounds for number fields: For non-CM number fields $K$, the regulator $R$ satisfies a lower bound of the form $R \geq A (log \Delta_K)^B$ for some effective constants $A$ and $B$ depending on the signature of $K$. Combined with our existing discriminant completeness bounds, this yields completeness guarantees up to an explicit regulator bound for certain signatures (though this would be a somewhat small bound for most signatures).

  To make this work within the NFBound class, I've added the regulator to display_reason, and have added a method clear_regulators which does essentially the same thing as clear_signatures, except for the regulator $R$ instead of the discriminant $D$. We then call self.clear_regulator as part of Completeness 1 in _one_n and return True if all signatures have been certified.

  https://beta.lmfdb.org/NumberField/?signature=%282%2C0%29&regulator=0-7
  http://localhost:37777/NumberField/?signature=%282%2C0%29&regulator=0-7

  https://beta.lmfdb.org/NumberField/?signature=%282%2C1%29&regulator=0-0.5
  http://localhost:37777/NumberField/?signature=%282%2C1%29&regulator=0-0.5

  https://beta.lmfdb.org/NumberField/?signature=%285%2C1%29&regulator=0-6
  http://localhost:37777/NumberField/?signature=%285%2C1%29&regulator=0-6

I've also been adding various extra comments and docstrings throughout the completeness.py file, mainly for my own understanding of how everything works. I thought I'll keep them in as it might benefit others (but can also take them out if you'd think they're unnecessary).

[roed314]

I need to focus on final exams for a few days, but I'll be happy to look this once I'm done!

[rvisser7]

Thanks - of course no rush! I've realised there are many more cases we could still add, though I think the above is probably enough for this PR. :)

I've opened a separate issue #7017 to help track more cases we could add to the completeness checker in the future.