Superspecial plane quintics with large automorphism groups

In this paper, we study plane quintic curves whose automorphism groups have order greater than 10, as well as those with cyclic automorphism groups of order 8 and 10. The latter two cases are represented as one-parameter families, where their superspeciality can be explicitly described in terms of a truncation of certain Gaussian hypergeometric series. Applying this characterization, we determine the exact number of isomorphism classes of superspecial plane quintic curves with automorphism groups $\cong \mathbb{Z}\mathrm{/}10\mathbb{Z}$. We also provide an efficient algorithm to enumerate such curves with automorphism groups $\cong \mathbb{Z}\mathrm{/}8\mathbb{Z}$, and provide the computational results for the range $13<p<10000$.