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

TL;DR

This paper determines the twisting Sato-Tate group of a specific genus 3 hyperelliptic curve, constructs explicit twists realizing all subgroup types, and proves the generalized Sato-Tate conjecture for their Jacobians.

Contribution

It explicitly computes the twisting Sato-Tate group for the curve and verifies the generalized Sato-Tate conjecture for all its twists.

Findings

All subgroup types of the twisting Sato-Tate group are realized by explicit twists.

The generalized Sato-Tate conjecture is proven for Jacobians of all Q-twists of the curve.

The structure of the twisting Sato-Tate group is fully characterized.

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 $y^2 = x^{8} - 14x^4 + 1$.