Twists of the genus 2 curve Y2 = X6 + 1

Abstract

Here we study the twists of the genus 2 curve given by the hyperelliptic equation Y2=X6+1 over any field of characteristic different from 2, 3 or 5. Since any curve of genus 2 with group of automorphisms of order 24 is isomorphic (over an algebraically closed field) to the given one, the study of this set of twists is equivalent to the classification, up to isomorphisms defined over the base field, of curves of genus 2 with that number of automorphisms. This contribution closes the series of articles on the classification of twists of curves of genus 2. The knowledge of these twists can be of interest in a wide range of arithmetical questions, such as the Sato-Tate or the Strong Lang conjectures among others.