Lifting low-gonal curves for use in Tuitman's algorithm

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