Leave $ A $ be an abelian variety of dimension $ g $ defined on a numeric field $ K $. Suppose $ A $ It has a main polarization and $ ell $ It's a prime number. We have a pair of Weil:

$$

e_ ell: A[ell] times A[ell] rightarrow mu_ ell

$$

I do not know if Weil's pairing is unique, so my question is this:

Given a subgroup $ G subset A[ell]$ of order $ ell ^ 2 $, there's a couple from Weil $ e_ ell $ (possibly depending on $ G $) such that there are $ P, Q in G $ Y $ e_ ell (P, Q) $ It is a primitive $ ell $-the root of the unit?

I know by here that for any point. $ P in G $, there is a point $ Q in A[ell]$ such that $ e_ ell (P, Q) $ It is a primitive $ ell $-the root of the unit. Thanks in advance for the comments and answers.