Some Remarks on the Hyperelliptic Moduli of Genus 3
In 1967, Shioda  determined the ring of invariants of binary octavics and their syzygies using the symbolic method. We discover that the syzygies determined in  are incorrect. In this paper, we compute the correct equations among the invariants of the binary octavics and give necessary and sufficient conditions for two genus 3 hyperelliptic curves to be isomorphic over an algebraically closed field k, char k ≠ 2, 3, 5, 7. For the first time, an explicit equation of the hyperelliptic moduli for genus 3 is computed in terms of absolute invariants.