Effective Artin-Schreier-Witt theory for curves

We present an algorithm which, given a connected smooth projective curve $X$ over an algebraically closed field of characteristic $p>0$ and its Hasse–Witt matrix, as well as a positive integer $n$, computes all étale Galois covers of $X$ with group $\mathbb{Z}\mathrm{/}{p}^n\mathbb{Z}$. We compute the complexity of this algorithm when $X$ is defined over a finite field, and provide a complete implementation in SageMath, as well as some explicit examples. We then apply this algorithm to the computation of the cohomology complex of a locally constant sheaf of $\mathbb{Z}\mathrm{/}{p}^n\mathbb{Z}$-modules on such a curve.