[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
[Tables]
[FAQ] -
[Statistics] -
MAGMA: [Functions] [Data] -
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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 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.
We present complete tables of all congruence subgroups of PSL(2,Z) 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 1A^{0} 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) instead of "Γ bar". |
Index | The index of the group G in PSL(2,Z). |
con | The number of conjugates of G under outer automorphisms. |
len | The number of PSL(2,Z) conjugates of G. |
c_{2} | The number of classes of elements of order 2. |
c_{3} | 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 of the PSL(2,Z) conjugates of G 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 G. This is also written in partition notation with one part for each PSL(2,Z) conjugate of G. This data gives information on the degree of the field generated by a `minimal' field of automorphic functions of G. |
Supergroups | The list of all direct supergroups V of the group G with links to the groups V. That is, all subgroups G of PSL(2,Z) such that V is a maximal proper subgroup of G (up to PGL(2,Z) conjugacy). |
Subgroups | The list of all proper maximal subgroups of G of genus less than or equal to 24 with links to the groups. |
Matrix Generators |
The images of the generators of G in 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".
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.
Why is group X not in the tables ?
The first point is that we only look for groups in SL(2,Z) which contain the center (i.e. ± I). The reason being that we are really only interested in groups acting on the upper half plane, and when we look at preimages in SL(2,Z) of such groups these contain the center. Equivalently, we are only listing subgroups of PSL(2,Z) = SL(2,Z)/<-I>. So if a subgroup of SL(2,Z) does not contain the centre, it is not listed in our tables. Our naming convention in the tables is that G always means ± G (= group generated by G and -Identity). For example Γ(3) is not in the list. The Γ(3) which appears in the "name" column for 3D^{0} means that 3D^{0} is ±Γ(3) (in our paper we use the notation "Γ(3) bar" for this group).Another reason a group might not be immediately recognizable is that we only list the groups up to outer automorphisms. The inner automorphisms of SL(2,Z) have index 2 in the the outer automorphisms. The outer automorphisms are generated by the inner automorphisms together with x → t^{-1}xt, where t is the matrix [-1,0,0,1].
Of course it is also possible that there is a mistake in the tables! If you think there is an error, please let us know.
What are the "matrix generators" listed ?
The generators are the generators of the image of the group G in SL(2,Z/NZ) where N is the level of G. If G has level N, then it contains Γ(N) (no ± !) which is the kernel of reduction mod N. So giving these generators in SL(2,Z/NZ) completely specifies the group. The group is generated by lifts of these matrices to SL(2,Z) together with generators of Γ(N).Isn't it possible to find "nicer" generators ?
Yes. We are working on this and hope to update the tables.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.
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.