Consider a smooth projective curve $\overline{ C}$ over a finite field $\mathbb{ F}_q$, equipped with a simply branched morphism $\overline{ C} \to \mathbb{ P}^1$ of degree $d \leq 5$. Assume char$\, \mathbb{ F}_q > 2$ if $d \leq 4$, and char$\, \mathbb{ F}_q > 3$ if $d=5$. In this paper we describe how to efficiently compute a lift of $\overline{ C}$ to characteristic zero, such that it can be fed as input to Tuitman's algorithm for computing the Hasse-Weil zeta function of $\overline{ C} / \mathbb{ F}_q$. Our method relies on the parametrizations of low rank rings due to Delone-Faddeev and Bhargava.
Categories: math.NT