Superspecial genus-$4$ double covers of elliptic curves

Abstract: In this paper we study genus-$4$ curves obtained as double covers of elliptic curves. Firstly we shall give explicit defining equations of such curves with explicit criterion for whether it is nonsingular, and show the irreducibility of the long polynomial determining whether the genus-4 curve is nonsingular or not, in any characteristic $\ne 2,3$. Secondly, as an application, we enumerate superspecial genus-$4$ double covers of elliptic curves in small characteristic.