An earlier preprint version is also available.
A Howe curve (so named by Kudo, Harashita, and Senda in earlier work) is a curve of genus 4 obtained as the fiber product of two genus-1 double covers of P1. In this paper, we present a simple algorithm for testing isomorphism of Howe curves, and we propose two main algorithms for finding and enumerating these curves: One involves solving multivariate systems coming from Cartier–Manin matrices, while the other uses Richelot isogenies of curves of genus 2. Comparing the two algorithms by implementation and by complexity analyses, we conclude that the latter enumerates curves more efficiently. Using these algorithms, we show that there exist superspecial curves of genus 4 in characteristic p for every prime p with 7 < p < 20000.
We have Magma code available for both enumeration algorithms, as well as Magma code required for a computation that is used in the proof of one of our theorems. The code is officially hosted as supplementary material on the publisher's web site, but you can also find the code used for the second enumeration algorithm here:
(I used this type of curve in my paper Quickly constructing curves of genus 4 with many points, and the algorithm implemented in the file Superspecial.magma is quite similar to the algorithms in this earlier paper.)