DECOMPOSITION OF THE JACOBIAN OF SOME TWISTS OF A GENUS $2$ CURVE

Cardona and Lario [‘Twists of the genus 2 curve $y^2 = x^6+1$ ’, J. Number Theory 209 (2020), 195–211] gave a complete classification of the twists of the curve $y^2 = x^6+1$ . In this paper, we study the twists of the curve whose automorphism group is defined over a biquadratic extension of the rationals. If the twists are of type B or C in the Cardona–Lario classification, we find a pair of elliptic curves whose product is isogenous with the Jacobian of the twist.