base-change of elliptic curves over Q

[AurelPage]

From /roadmap: Base change: the user could select an extension field and see the page for the base-changed curve. This would be easy for cases where the base-change curve is already in the database.

Currently, from elliptic curves over number fields that are base-change, there is a link to the curve over Q. This asks for a link in the opposite direction.

[JohnCremona]

We don't want to allow users to create arbitrary base-changes, and I think it would be too expensive to do anything like this on the fly. One possibility is to do a one-off computation which went through all curves E/Q in the database, and all number fields F for which we have any curves, compute the base-change, see if that is in the database and if so store its label. Or use the fact that every E/F already knows if it s base-change (see example below). Then every E/Q would have stored a list of labels of its base-changes in the database. To keep this up-to-date, two things need to be done: (1) when extra curves /Q are added, this procedure needs to be done for them; (2) when extra curves over fields F/Q are added which are base-change (something which is known at upload time) the appropriate ec/Q records could be updated.

This would be quite a lot of work.

Then the home page of any E/Q which has any base-changes in the database could show them. For example, there are 44 different base-changes of 11a1:

sage: len(list(db.ec_nfcurves.search({'base_change':{'$contains':'11.a1'}})))
44

[JohnCremona]

In fact it turns out to be rather easy to get a list, for each EC/Q, of which EC/F are its base-changes in the database. Doing this for 1 curve from each isogeny class reveals that out of 1.74 million isogeny classes, only 5221 have base-changes present. A simple rewrite (after creation of a new column) would record these lists for all curves.