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

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

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...

Cheers,
Bill.