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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 to classify all group actions on Riemann Surfaces [MR:1796706] . 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] , 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/] .

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/] .