Enumeration Algorithm for Genus-4 Superspecial Hyperelliptic Curves with Automorphism Group $Q_8$

In this paper, we propose an algorithm to enumerate genus-4 superspecial hyperelliptic curves whose automorphism groups isomorphic to the quaternion group. By implementing this algorithm with Magma, we successfully obtain the number of isomorphism classes of such curves in every characteristic $7\le p<10000$. Interestingly, the experimental results lead us to the conjecture that there exist exactly $[p\mathrm{/}48]$ isomorphism classes of such curves if $p\equiv 1,7(\mathrm{m}\mathrm{o}\mathrm{d}8)$, whereas such curves exist if $p\equiv 3,5(\mathrm{m}\mathrm{o}\mathrm{d}8)$