Algorithms to enumerate superspecial Howe curves of genus 4

TL;DR

This paper introduces algorithms for enumerating superspecial Howe curves of genus 4, compares their efficiency, and demonstrates the existence of such curves in various characteristics.

Contribution

It presents two novel algorithms for finding and enumerating Howe curves, with a detailed complexity analysis and practical implementation results.

Findings

The Richelot isogeny-based algorithm is more efficient for enumeration.

Superspecial Howe curves of genus 4 exist for all primes between 8 and 20000.

The paper provides a practical approach to classify these curves in characteristic p.

Abstract

A Howe curve is a curve of genus $4$ obtained as the fiber product of two genus-$1$ double covers of $\mathbf{P}^1$. 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 $7 < p < 20000$.