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Let $X$ be a compact Riemann surface (equivalently, a smooth projective curve over $\C$), leg of genus $g$, let $G$ be a group of automorphisms acting on $X$, and let $g_0$ be the genus of the quotient $Y:=X/G$. The natural projection $X \to Y$ is branched at $r$ points in $Y$, and the corresponding elliptic generators of the monodromy group have order $m_1$, $m_2$, $\ldots$, $m_r$; the sequence of integers $[g_0; m_1, \ldots, m_r]$ is called the signature of the group action.

The label for the family of higher genus curves with a group $G \simeq$ SmallGroup$(n,d)$ acting on it with signature $[g_0; m_1, \ldots, m_r]$ is given as $$g.n-d.g_0.m_1-m_2- \cdots -m_r$$ For example, the genus 3 Hurwitz curve with automorphism group PSL$(2,7) \cong $SmallGroup$(168,42)$ and signature $[0;2,3,7]$ is labeled: $$5.168-42.0.2-3-7$$ There may be several inequivalent actions described by that label, though. We also distinguish the actions by which conjugacy classes in $G$ the monodromy generators are from, creating passport labels. For our previous example $$5.168-42.0.2-3-7.1 \text{ and } 5.168-42.0.2-3-7.2$$ represent the two distinct actions of PSL$(2,7)$ as a Hurwitz group on a genus $3$ curve up to refined passports.

The numbers $1$ and $2$ are assigned based on the increasing order of the numbers labeling the corresponding conjugacy classes in Magma when the data is initially produced. At that point, we use a particular function to convert a group of the form SmallGroup$(n,d)$ to a subgroup of $S_{|G|}$, the symmetric group on $|G|$ elements. The conjugacy class numbers in the data correspond to this particular permutation group in Magma.