Abstract

A Howe curve is a curve of genus $4$ obtained as the fiber product of two genus-$1$ double covers of $P^1$. We present a simple algorithm for testing isomorphism of Howe curves, and we propose two main algorithms for finding and enumerating superspecial Howe 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 superspecial Howe curves more efficiently. Using these algorithms, we show that there exist superspecial curves of genus $4$ in characteristic $p$ for every prime $p$ with $7<p<20000$.

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Mathematical Subject Classification

Supplementary material

Algorithms for enumerating superspecial Howe curves

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