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].

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.