Several problems in arithmetic geometry reduce to computing the monodromy group (or, closely related, the image of the Galois representation) of a suitable abelian variety A over a number field k. On the one hand, the Mumford-Tate conjecture describes the neutral connected component of the monodromy group. On the other hand, the group of components of the monodromy group is naturally isomorphic to the Galois group of a finite extension \(k(e_A)/k\), which encodes significant arithmetic information. For instance, the extension \(k(e_A)/k(EndA)\) detects exceptional Tate classes on A. However, no general algorithm to compute \(k(e_A)/k\) is currently known, even conjecturally. In this talk, we compute this invariant for a family of degenerate abelian varieties: the Jacobians \(J_m\) of curves defined by the equation \(y^2 = x^m + 1\) over \(k=Q\). Our strategy is to identify exceptional Tate classes on \(J_m\) using CM theory, relate them to Tate classes on a suitable Fermat variety, and apply a result of Deligne to determine the field of definition of these classes.