# Higher genus curve downloaded from the LMFDB on 12 March 2026. # Search link: https://www.lmfdb.org/HigherGenus/C/Aut/?genus=4&group=[12,5] # Query "{'group': '[12,5]', 'genus': 4, 'cc.1': 1}" returned 8 refined passports, sorted by genus. # Each entry in the following data list has the form: # [Refined passport label, Genus, Group, Group order, Dimension, Signature] # For more details, see the definitions at the bottom of the file. "4.12-5.0.6-6-6.2" 4 "[12,5]" 12 0 "[0,6,6,6]" "4.12-5.0.6-6-6.1" 4 "[12,5]" 12 0 "[0,6,6,6]" "4.12-5.0.2-2-3-6.3" 4 "[12,5]" 12 1 "[0,2,2,3,6]" "4.12-5.0.2-2-3-6.4" 4 "[12,5]" 12 1 "[0,2,2,3,6]" "4.12-5.0.2-2-3-6.1" 4 "[12,5]" 12 1 "[0,2,2,3,6]" "4.12-5.0.2-2-3-6.6" 4 "[12,5]" 12 1 "[0,2,2,3,6]" "4.12-5.0.2-2-3-6.5" 4 "[12,5]" 12 1 "[0,2,2,3,6]" "4.12-5.0.2-2-3-6.2" 4 "[12,5]" 12 1 "[0,2,2,3,6]" #Refined passport label (passport_label) -- # Let $X$ be a compact Riemann surface (equivalently, a smooth projective curve over $\C$) 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 generators of the monodromy group have orders $m_1$, $m_2$, $\ldots$, $m_r$; 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 # $$g.n\text{-}d.g_0.m_1\text{-}m_2\text{-} \cdots \text{-}m_r$$ # For example, the genus 3 Hurwitz curve with automorphism group PSL$(2,7) \simeq $SmallGroup$(168,42)$ and signature $[0;2,3,7]$ is labeled: # $$\text{5.168-42.0.2-3-7}$$ # 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 # $$\text{5.168-42.0.2-3-7.1} \text{ and } \text{5.168-42.0.2-3-7.2}$$ # represent the two distinct actions of PSL$(2,7)$ as a Hurwitz group on a genus $3$ curve up to refined passports. # The suffixes $1$ and $2$ are ordinals that are assigned by lexicographically ordering the sequence of conjugacy class identifiers associated to a refined passport. # In order to explicitly identify elements of $G$ listed in generating vectors for a given refined passport (and elsewhere), we choose a particular permutation representation of $G$ as a subgroup of $S_{|G|}$, the symmetric group on $|G|$ elements (this choice depends only on the isomorphism class of $G$ and is the same for all $G$ with the same group identifier). # Genus -- # The **genus** of a smooth projective geometrically integral curve $C$ defined over a field $k$ is the dimension of the $k$-vector space of regular differentials $H^0(C, \omega_C)$. When $k=\C$ this coincides with the topological genus of the corresponding Riemann surface. # The quantity defined above is sometimes also called the **algebraic genus** or the **geometric genus** of $C$. Because of our assumption on the smoothness of $C$, it coincides with the **arithmetic genus** $H^1(C,\mathcal{O}_C)$. # Group -- # The computer algebra systems GAP and Magma both include a database of small groups that includes all groups of order up to 2000, except for groups of order 1024. Groups in this database are identified by an ordered pair $[n,d]$ where $n$ is the order of the group and $d$ is a positive integer that distinguishes the group from others of the same order (the value of $d$ is the same in both GAP and Magma). # In both systems, the command "SmallGroup($n$,$d$)" will return an explicit representation of the group in terms of generators and relations. # Magma also associates a descriptive name to each small group. For example, the command "GroupName(SmallGroup(24,3))" returns the string "SL(2,3)". #Group order (group_order) -- # The **order** of a group is its cardinality as a set. #Dimension (dim) -- # Given a group of automorphisms $G$ acting on a curve $X/\C$ of genus $g$, with signature $[g_0;m_1, \ldots, m_r]$, the **dimension** of the family of curves with this signature is $3g_0+r-3$, where $r$ is the number of branch points of the map $X \to X/G$ and $g_0$ is the quotient genus. # Signature -- # Let $G$ be a group of automorphisms acting on a curve $X/\C$ of genus at least $2$, let $g_0$ be the genus of the quotient $Y:=X/G$, and let $B$ be the set of branch points of the projection $\phi\colon X\to Y$. For each standard generator for $\pi_1(Y-B,y_0)$ corresponding to the elements of $B=\{y_1,\ldots,y_r\}$ and a fixed pre-image of $y_0$, there is a lift of that loop to a path in $X - \phi^{-1}(B)$ starting at the particular pre-image of $y_0$. The lifted path ends at some (possibly different) pre-image of $y_0$. Varying the particular pre-images of $y_0$ and then recording the end point corresponding to that starting point induces a permutation on those pre-images. # We define $m_1 \leq m_2 \leq \ldots \leq m_r$ to be the orders of the permutations described above, one for each of the $r$ standard generators. # The sequence of integers $[g_0; m_1, \ldots, m_r]$ is the **signature** of the group action.