  Let $\mathcal{X}$ be an algebraic curve of genus $g$ defined over an
algebraically closed field $K$ of characteristic $p \geq 0$, and $q$ a prime
dividing $|\mbox{Aut}(\mathcal{X})|$. We say that $\mathcal{X}$ is a $q$-curve.
Homma proved that either $q \leq g+1$ or $q = 2g+1$, and classified
$(2g+1)$-curves. In this note, we classify $(g+1)$-curves, and fully
characterize the automorphism groups of $q$-curves for $q= 2g+1, g+1$. We also
give some partial results on $q$-curves for $q = g, g-1$.
