ECQ: show which of a curve's base-changes are in the database, if any

[JohnCremona]

See #3837 for related discussion.

On each home page of an elliptic curve over Q in the Related Objects box we should list the labels of (as links) all of its base-change curves in the ECNF database. This requires no database changes, as the ec_nfcurves table has a column 'base_change' which lists the labels of any curves /Q of which it is the base change, and these are listed in its Related Objects box.

[JohnCremona]

See various comments on #6834 which are relevant here, specifically that the ec_nfcurves table has a column 'base_change' which is a (possibly empty) list of labels of all the curves (over Q or an intermediate field) of which the curve is a base-change, but that the labels are cometimes Clabels and sometimes lmfdb_labels.

Pending a revision of that column in ec_nfcurves (which I will do) I have implemented this feature, and will make a PR for it to allow people to see how it looks, as there is more than one possibility: (1, just implemented) listing all the ecnf labels in the Related Objects box (each a link to that curve's home page) under a subheading "Base changes in the database". Other possibilities include (2) having one entry in the Related Objects box "Base changes in the database", being a link to a list of these curves similar to a search results page; (3) not using the Related Objects box but instead having a section on the main page with a list of the b-c's in the database.

[AndrewVSutherland]

I think I would vote for (2), as this feels similar to twists, which are already handled this way. But I'm wondering what this includes? Ideally I think it would include all curves other than this one that are base changes of (any) model of E over its minimal field of definition, so if E/K is the base change of an elliptic curve E_0 over Q the link would include all bases changes of E_0, even though they are not, stricitly speaking base changes of E. But in cases where E/K is the base change of two different E_0 (that are not isomorphic over Q but are over K), this might be tricky to implement? But maybe I am over-thinking this. What exactly do you have in mind?

[JohnCremona]

Thanks for the comments -- I hope you are overthinking this, as I am finding it hard to follow you. Here's what I have implemented (in my branch ecq_show_bc): for every E0/Q on its home page we list (and link to) all E/K for fields K bigger than Q such that E is the base-change from Q to K of E0.

I have not implemented showing anything more on an ECNF curve home page than what we already show, which is, for E/K, a list of curves E0/K0 where K0 is a proper subfield of K and E is the b.c. from K0 to K of E0.

Perhaps you also want to show on an ECNF homepage of E/K any curves E1/K1 where there is a common subfield K0 of both K and K1 and a curve E0/K0 such that both E and E1/K1 are b.c. of E0 from K0 to K (resp. K1)? That had not occurred to me. Users can do this in two steps anyway, at least when K0=Q.

I am about to make a PR with what I have done, which implements both (1) and (2) simultaneously, though currently (2) does not work properly -- see the PR, I am sure that an expert can fix it.