[Introduction] - Groups of Genus: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Congruence Subgroups of PSL(2,Z)

[Tables] - [Statistics] - MAGMA: [Functions] [Data] - Mirrors: [Berlin] [Montreal]

^ Introduction

The group Γ = PSL(2,Z) = SL(2,Z)/{ -1 } acts on the extended upper half plane H (the upper complex half plane extended by the rational numbers and infinity) by fractional linear transformations. The genus of a subgroup U of Γ is the genus of the corresponding surface H/U. The principal congruence subgroup of level N, Γ(N), is the image in PSL(2,Z) of the group {[a,b,c,d] in SL(2,Z) with [a,b,c,d] = [1,0,0,1] mod N}. A subgroup of Γ which contains some principal congruence subgroup is called a congruence subgroup. The level of a congruence subgroup U is the smallest N such that Γ(N) is a subgroup of U.

^ The Tables

We present complete tables of all congruence subgroups of PSL(2,Z) of of genus 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, and 24. In the tables we use the notation (level)(label)(genus) to label the subgroups. So for example 1A0 is the name of PSL(2,Z). The columns of the tables contain the following data.
Name We list standard names of some of the groups. Due to the restrictions of HTML we write Γ for PSL(2,Z).
Index The index of the group in PSL(2,Z).
con The number of conjugates under outer automorphisms.
len The number of PSL(2,Z) conjugates.
c2 The number of classes of elements of order 2.
c3 The number of classes of elements of order 3.
Cusps The cusp widths written in partition notation.
Gal The length of the orbits of the images of each the PSL(2,Z) conjugates of the subgroup U in PGL(2,Z/mZ) under conjugation by the subgroup D consisting of the matrices [1,0,0,x] where x is in (Z/mZ)* and m is the level of U. This is also written in partition notation with one part for each PSL(2,Z) conjugate of U. This data gives information on the degree of the field generated by a `minimal' field of automorphic functions of U.
Supergroups The list of all direct supergroups V of the group U with links to the groups V. That is, all subgroups U of PSL(2,Z) such that V is a maximal proper subgroup of U (up to PGL(2,Z) conjugacy).
Subgroups The list of all proper maximal subgroups of genus less than or equal to 24 with links to the groups.
The generators of G over SL(2,Z/mZ) as matrices, where m denotes the level of G.

Clicking on a "^" in the left margin takes you back to the top of the table. Clicking on a ">" switches between "Cusps, Gal, Supergroups, Subgroups" and "Matrix Generators".

Valid HTML 4.01!

Warning: Some browsers may have difficulties jumping to a specific group in the tables. In some cases the browser will jump to a specific group, if the corresponding page has been loaded before. We apologize for any inconvenience this may cause.

^ MAGMA Functions

The files pre.m, csg.m, func.m, table.m, and html.m in the archive csg.m.tar.gz contain the functions we used to compute the tables of congruence subgroups. Some of the functions are described in README.txt.


^ MAGMA Data

The files in the archive below contain the data of all congruence subgroups of genus less than 25. MAGMA has difficulties reading in larger textfiles, so we decided to split up the data. csgN-levM.dat contains the data of group of genus N and level greater than or equal to M. Before you type 'load "csgN.dat"' in MAGMA at least csg.m (which loads pre.m) should be loaded, otherwise MAGMA does not know about the format of the data. The file csgN-levM.dat also reads the files csgK-levL.dat, where csgK-levL.dat contains data of lower genus and/or level. Loading csg24.dat gives you all groups from the tables -- this may take some time. The list of congruence subgroups is called L.


Chris Cummins and Sebastian Pauli