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

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

arXiv comment34 pages, comments welcome

References in corpus (4)

• Sato-Tate distributions and Galois endomorphism modules in genus 2
• Computing Hasse-Witt matrices of hyperelliptic curves in average polynomial time
• Sato-Tate groups of some weight 3 motives
• Hyperelliptic curves of genus 3 with prescribed automorphism group

Cited by in corpus (2)

• Sato-Tate groups of abelian threefolds: a preview of the classification
• Computing twists of hyperelliptic curves