Degeneracy and Sato-Tate groups of $y^2=x^{p^2}-1$

We say that an abelian variety is degenerate if its Hodge ring is not generated by divisor classes. Degeneracy leads to some interesting challenges when computing Sato-Tate groups, and there are currently few examples and techniques presented in the literature. In this paper we focus on the Jacobians of the family of curves $C_{p^2}:y^2=x^{p^2}-1$, where $p$ is an odd prime. Using a construction developed by Shioda in the 1980s, we are able to characterize so-called indecomposable Hodge classes as well as the Sato-Tate groups of these Jacobian varieties. Our work is inspired by computation, and examples and methods are described throughout the paper.