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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 \(y^2 = x^{8} - 14x^4 + 1\).

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