A paper with a detailed description of the data may be found at [ www.math.grinnell.edu/~paulhusj/Paulhus-lmfdb.pdf ].

### Original Data of Thomas Breuer

The basis for this data is the work of Thomas Breuer [MR:1796706 ] to classify all group actions on Riemann surfaces. Breuer wrote code to implement his algorithm, and ran that code in GAP up to genus 48. Using his data, and a slightly modified version of his code in Magma, given a group and a signature, we compute generating vectors for each list of conjugacy classes in the group which correspond to the data in the signature.

### Full Automorphism Groups

An algorithm written by Jennifer Paulhus, based on work of John F. X. Ries [MR:1097170 ], then determines whether the action (given as a generating vector) is the full action for that family in the moduli space. If not, the larger group which is the full automorphism group for a generic curve in the family is also computed. The code may be found at [github.com/jenpaulhus/group-actions-RS ].

### Additional Data

**Jacobian variety decomposition** These decompositions, using the group action, are based on work described in [MR:2403651
, arXiv:math/0312284
], and code found at [github.com/jenpaulhus/decompose-jacobians
]. The factors are notated with either an $E$ for an elliptic curve, or an $A_d$ for an abelian variety of dimension $d$.

**Corresponding Character(s)** This entry is a list of positive integers (one for each factor in the Jacobian variety decomposition) enumerating irreducible $\mathbb{C}$-characters from the character table of the group (as listed in Magma). Each factor in the Jacobian variety decomposition corresponds to a particular irreducible $\mathbb{Q}$-character of the group, so these numbers each represent an irreducible $\mathbb{C}$-character whose Galois orbit is the corresponding $\mathbb{Q}$-character for that factor. A word of caution: the numbers in Magma may vary depending on how the group is represented. The numbers in this entry are for the permutation group generated by the listed generating vectors.

**Dimension of the corresponding Shimura variety** For their paper [MR:3455876
], Paola Frediani, Alessandro Ghigi, and Matteo Penegini wrote code which computes this dimension. That code may be found at [www.dima.unige.it/~penegini/publications/PossGruppigFix_v2Hwr.m
].

**Hyperelliptic or Cyclic Trigonal Curves** Whether an action defines a hyperelliptic curve or a cyclic trigonal curve is determined using code written by David Swinarski [faculty.fordham.edu/dswinarski/RiemannSurfaceAutomorphisms/
] which is described in his paper [arXiv:1607.04778
].

**Equations** Equations of hyperelliptic curves are from [MR:2035219
]. Equations for all genus 3 curves with automorphisms come from [MR:1954371
]. And equations for curves with large automorphisms (i.e. greater than 4(g-1) ) are found in [faculty.fordham.edu/dswinarski/RiemannSurfaceAutomorphisms/
].

## Additional acknowledgments

Jen Paulhus, using group and signature data originally computed by Thomas Breuer.