Abstract Here we study the twists of the genus 2 curve given by the hyperelliptic equation Y 2 = X 6 + 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.

中文翻译：

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属 2 曲线的扭曲 Y2 = X6 + 1

摘要 这里我们研究由超椭圆方程 Y 2 = X 6 + 1 给出的属 2 曲线在特征不同于 2、3 或 5 的任何场上的扭曲。由于具有 24 阶自同构群的属 2 曲线是同构（在代数闭域上）到给定的，对这组扭曲的研究等效于分类，直到定义在基域上的同构，属 2 的曲线与该数量的自同构。这一贡献结束了关于属 2 曲线扭曲分类的系列文章。这些扭曲的知识可能对广泛的算术问题感兴趣，例如 Sato-Tate 或 Strong Lang 猜想等。