  Let $E_\lambda$ be the Legendre family of elliptic curves. Given $n$ linearly
independent points $P_1,\dots , P_n \in
E_\lambda\left(\overline{\mathbb{Q}(\lambda)}\right)$ we prove that there are
at most finitely many complex numbers $\lambda_0$ such that $E_{\lambda_0} $
has complex multiplication and $P_1(\lambda_0), \dots ,P_n(\lambda_0)$ are
dependent over $End(E_{\lambda_0})$. This implies a positive answer to a
question of Bertrand and, combined with a previous work in collaboration with
Capuano, proves the Zilber-Pink conjecture for a curve in a fibered power of an
elliptic scheme when everything is defined over $\overline{\mathbb{Q}}$.
