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Published online by Cambridge University Press: 20 November 2018
Zarhin proves that if $C$ is the curve ${{y}^{2}}\,=\,f(x)$ where $\text{Ga}{{\text{l}}_{\mathbb{Q}}}(f(x))\,=\,{{S}_{n}}$ or ${{A}_{n}}$, then $\text{En}{{\text{d}}_{\overline{\mathbb{Q}}}}(J)\,=\,\mathbb{Z}$. In seeking to examine his result in the genus $g\,=\,2$ case supposing other Galois groups, we calculate $\text{En}{{\text{d}}_{\overline{\mathbb{Q}}}}(J)\,{{\otimes }_{\mathbb{Z}}}\,{{\mathbb{F}}_{2}}$ for a genus 2 curve where $f(x)$ is irreducible. In particular, we show that unless the Galois group is ${{S}_{5}}$ or ${{A}_{5}}$, the Galois group does not determine $\text{En}{{\text{d}}_{\overline{\mathbb{Q}}}}(J)$.