Computing superspecial hyperelliptic curves of genus 4 with automorphism group properly containing the Klein 4-group

In algebraic geometry or number theory, enumerating or finding superspecial curves in positive characteristic p is important both in theory and in computation. In this paper, we propose feasible algorithms to enumerate or find superspecial hyperelliptic curves of genus 4 with automorphism group properly containing the Klein 4-group. By executing the algorithms on Magma, we succeeded in enumerating such superspecial curves for all primes p with 19≤p<500, and in finding a single one for all primes p with 19≤p<7000.