Re: question on possibly extending the ell2cover(E) command

[Aurel Page <aurel.page@normalesup.org>]

On 10/21/26 18:10, Bill Allombert wrote:
> On Tue, Oct 20, 2026 at 01:18:51PM -0700, American Citizen wrote:
> > We read:
> >
> > ? ?ell2cover
> > ell2cover(E): if E is an elliptic curve over Q, returns a basis of the set
> > of everywhere locally soluble 2-covers of the curve E. For each
> > cover a pair [R,P] is returned where y^2-R(x) is a quartic curve and P
> > belongs to E(k), where k = Q(x)[y] / (y^2-R(x)).
> >
> > Is there a possible way to extend this command so that it could be modified
> > by known independent points on the curve, ie. only provide two covers which
> > might contain the other unknown points, i.e.
> >
> > ell2cover(E,P) returns a set of everywhere locally soluble 2-covers which
> > are not included in the coverage of independent points P
> The real problem is that the total number of 2-cover is equal 2^r where r
> is the Selmer rank and this can get large very quickly...
How about returning a basis of some (arbitrary, or maybe random) 
F2-complement of the F2-vector space generated by known points?

Cheers,
Aurel