We give a new algorithm for constructing Picard curves over a finite field with a given endomorphism ring. This has important applications in cryptography since curves of genus 3 allow one to work over smaller fields than the elliptic curve case. For a sextic CM-field K containing the cube roots of unity, we define and compute certain class polynomials modulo small primes and then use the Chinese Remainder Theorem to construct the class polynomials over the rationals. We also give some examples.

We determine the twisting Sato-Tate group of the genus 3 hyperelliptic curve y 2 = x 8 − 14x 4 + 1 and show that all possible subgroups of the twisting Sato-Tate group arise as the Sato-Tate group of an explicit twist of y 2 = x 8 − 14x 4 + 1. Furthermore, we prove the generalized Sato-Tate conjecture for the Jacobians of all Q-twists of the curve y 2 = x 8 − 14x 4 + 1.

We determine the twisting Sato-Tate group of the genus 3 hyperelliptic curve y 2 = x 8 − 14x 4 + 1 and show that all possible subgroups of the twisting Sato-Tate group arise as the Sato-Tate group of an explicit twist of y 2 = x 8 − 14x 4 + 1. Furthermore, we prove the generalized Sato-Tate conjecture for the Jacobians of all Q-twists of the curve y 2 = x 8 − 14x 4 + 1.