# Higher genus curve downloaded from the LMFDB on 08 August 2024.
# Search link: https://www.lmfdb.org/HigherGenus/C/Aut/?genus=6&inc_hyper=only
# Query "{'genus': 6, 'hyperelliptic': True, 'cc.1': 1}" returned 43 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.
"6.2-1.0.2-2-2-2-2-2-2-2-2-2-2-2-2-2.1" 6 "[2,1]" 2 11 "[0,2,2,2,2,2,2,2,2,2,2,2,2,2,2]"
"6.4-1.0.2-2-2-2-2-2-4-4.1" 6 "[4,1]" 4 5 "[0,2,2,2,2,2,2,4,4]"
"6.4-2.0.2-2-2-2-2-2-2-2-2.10" 6 "[4,2]" 4 6 "[0,2,2,2,2,2,2,2,2,2]"
"6.4-2.0.2-2-2-2-2-2-2-2-2.7" 6 "[4,2]" 4 6 "[0,2,2,2,2,2,2,2,2,2]"
"6.4-2.0.2-2-2-2-2-2-2-2-2.1" 6 "[4,2]" 4 6 "[0,2,2,2,2,2,2,2,2,2]"
"6.6-2.0.2-2-2-2-6-6.1" 6 "[6,2]" 6 3 "[0,2,2,2,2,6,6]"
"6.8-1.0.2-2-2-8-8.2" 6 "[8,1]" 8 2 "[0,2,2,2,8,8]"
"6.8-4.0.2-2-4-4-4.1" 6 "[8,4]" 8 2 "[0,2,2,4,4,4]"
"6.8-1.0.2-2-2-8-8.1" 6 "[8,1]" 8 2 "[0,2,2,2,8,8]"
"6.8-3.0.2-2-2-2-2-4.1" 6 "[8,3]" 8 3 "[0,2,2,2,2,2,4]"
"6.12-1.0.2-4-4-6.1" 6 "[12,1]" 12 1 "[0,2,4,4,6]"
"6.12-4.0.2-2-2-2-6.1" 6 "[12,4]" 12 2 "[0,2,2,2,2,6]"
"6.16-8.0.2-2-4-8.2" 6 "[16,8]" 16 1 "[0,2,2,4,8]"
"6.16-8.0.2-2-4-8.1" 6 "[16,8]" 16 1 "[0,2,2,4,8]"
"6.24-6.0.2-2-2-12.2" 6 "[24,6]" 24 1 "[0,2,2,2,12]"
"6.24-6.0.2-2-2-12.1" 6 "[24,6]" 24 1 "[0,2,2,2,12]"
"6.26-2.0.2-13-26.7" 6 "[26,2]" 26 0 "[0,2,13,26]"
"6.26-2.0.2-13-26.2" 6 "[26,2]" 26 0 "[0,2,13,26]"
"6.26-2.0.2-13-26.4" 6 "[26,2]" 26 0 "[0,2,13,26]"
"6.26-2.0.2-13-26.5" 6 "[26,2]" 26 0 "[0,2,13,26]"
"6.26-2.0.2-13-26.6" 6 "[26,2]" 26 0 "[0,2,13,26]"
"6.26-2.0.2-13-26.9" 6 "[26,2]" 26 0 "[0,2,13,26]"
"6.26-2.0.2-13-26.10" 6 "[26,2]" 26 0 "[0,2,13,26]"
"6.26-2.0.2-13-26.11" 6 "[26,2]" 26 0 "[0,2,13,26]"
"6.26-2.0.2-13-26.12" 6 "[26,2]" 26 0 "[0,2,13,26]"
"6.26-2.0.2-13-26.1" 6 "[26,2]" 26 0 "[0,2,13,26]"
"6.26-2.0.2-13-26.3" 6 "[26,2]" 26 0 "[0,2,13,26]"
"6.26-2.0.2-13-26.8" 6 "[26,2]" 26 0 "[0,2,13,26]"
"6.28-3.0.2-2-2-7.3" 6 "[28,3]" 28 1 "[0,2,2,2,7]"
"6.28-3.0.2-2-2-7.1" 6 "[28,3]" 28 1 "[0,2,2,2,7]"
"6.28-3.0.2-2-2-7.2" 6 "[28,3]" 28 1 "[0,2,2,2,7]"
"6.48-6.0.2-4-24.4" 6 "[48,6]" 48 0 "[0,2,4,24]"
"6.48-6.0.2-4-24.1" 6 "[48,6]" 48 0 "[0,2,4,24]"
"6.48-6.0.2-4-24.3" 6 "[48,6]" 48 0 "[0,2,4,24]"
"6.48-29.0.2-6-8.2" 6 "[48,29]" 48 0 "[0,2,6,8]"
"6.48-29.0.2-6-8.1" 6 "[48,29]" 48 0 "[0,2,6,8]"
"6.48-6.0.2-4-24.2" 6 "[48,6]" 48 0 "[0,2,4,24]"
"6.56-7.0.2-4-14.5" 6 "[56,7]" 56 0 "[0,2,4,14]"
"6.56-7.0.2-4-14.3" 6 "[56,7]" 56 0 "[0,2,4,14]"
"6.56-7.0.2-4-14.2" 6 "[56,7]" 56 0 "[0,2,4,14]"
"6.56-7.0.2-4-14.4" 6 "[56,7]" 56 0 "[0,2,4,14]"
"6.56-7.0.2-4-14.6" 6 "[56,7]" 56 0 "[0,2,4,14]"
"6.56-7.0.2-4-14.1" 6 "[56,7]" 56 0 "[0,2,4,14]"
#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.