question on possibly extending the ell2cover(E) command

[American Citizen <website.reader3@gmail.com>]

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

Randall

p.s. This probably involved the generators of the 2-Selmer group