Sato-Tate Groups and Distributions of $y^{\mathrm{\ell }}=x(x^{\mathrm{\ell }}-1)$

Let $C_{\mathrm{\ell }}\mathrm{/}\mathbb{Q}$ denote the curve with affine model $y^{\mathrm{\ell }}=x(x^{\mathrm{\ell }}-1)$, where $\mathrm{\ell }\ge 3$ is prime. In this paper we study the limiting distributions of the normalized $L$-polynomials of the curves by computing their Sato-Tate groups and distributions. We also provide results for the number of points on the curves over finite fields, including a formula in terms of Jacobi sums when the field ${\mathbb{F}}_q$ satisfies $q\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}{\mathrm{\ell }}^2)$.