  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.
