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In this paper, we describe an algorithm that, for a smooth connected curve$X$ over a field $k$ with normal completion having arithmetic genus $p_a(X)$, afinite locally constant sheaf $\mathcal A$ on $X_{et}$ of abelian groups oftorsion invertible in $k$, represented by a smooth curve with normal completionhaving arithmetic genus $p_a(\mathcal A)$ and degree $n$ over $X$, computes thefirst \'etale cohomology $H^1(X_{k^{sep},et},\mathcal A)$ and the first \'etalecohomology with proper support $H^1_c(X_{k^{sep},et},\mathcal A)$ as sets oftorsors, in arithmetic complexity exponential in $n^{\log n}$, $p_a(X)$, and$p_a(\mathcal A)$. This is done via the computation of a groupoid schemeclassifying the relevant torsors (with extra rigidifying data).

Explicit computation of the first \'etale cohomology on curves