Consider a smooth projective curve
$\stackrel{\u0304}{C}$
over a finite field
${\mathbb{\mathbb{F}}}_{q}$,
equipped with a simply branched morphism
$\stackrel{\u0304}{C}\to {\mathbb{P}}^{1}$ of degree
$d\le 5$. Assume
$char{\mathbb{\mathbb{F}}}_{q}>2$ if
$d\le 4$, and
$char{\mathbb{\mathbb{F}}}_{q}>3$ if
$d=5$.
In this paper we describe how to efficiently compute a lift of
$\stackrel{\u0304}{C}$ to characteristic
zero, such that it can be fed as input to Tuitman’s algorithm for computing the Hasse–Weil zeta
function of
$\stackrel{\u0304}{C}\u2215{\mathbb{\mathbb{F}}}_{q}$.
Our method relies on the parametrizations of low rank rings due to Delone and
Faddeev, and Bhargava.
