The Sato-Tate conjecture for a Picard curve with complex multiplication (with an appendix by Francesc Fite)

Let C/Q be the genus 3 Picard curve given by the affine model y(3) = x(4) - x. In this paper we compute its Sato-Tate group, show the generalized Sato-Tate conjecture for C, and compute the statistical moments for the limiting distribution of the normalized local factors of C.