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. However, in order to say that the latter strategy outputs all superspecial Howe curves, we require a conjecture that all superspecial curves of genus 2 in characteristic p>2 are connected by a path of Richelot isogenies. Given a prime p, the algorithm verifies this conjecture before producing output.
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 for the first enumeration algorithm is available (or will be soon available) on Kudo's web site. The code for the second enumeration algorithm, and the code used in the proof, are 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.)