Connectedness extensions for abelian varieties

Suppose $A$ is an abelian variety over a field $F$, and $\ell$ is a prime not equal to the characteristic of $F$. Let $F_{\Phi,\ell}(A)$ denote the smallest extension of $F$ such that the Zariski closure of the image of the $\ell$-adic representation associated to $A$ is connected. Serre introduced this field, and proved that when $F$ is a finitely generated extension of ${\mathbf Q}$, $F_{\Phi,\ell}(A)$ does not depend on the choice of $\ell$. In this paper we study extensions $F_{\Phi,\ell}(B)/F$ for twists $B$ of a given abelian variety, especially when the abelian varieties are of Weil type.