The twisting Sato–Tate group of the curve $$y^2 = x^{8} - 14x^4 + 1$$ y 2 = x 8 - 14 x 4 + 1

Abstract

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 $$\mathbb Q$$ -twists of the curve $$y^2 = x^{8} - 14x^4 + 1$$ .