# Generators of $\Gamma(7)$, congruence subgroup of modular group

L.s.,

I try to do some calculations on the Klein Quarctic curve, but there is a basic thing I don't know how to compute.

Let $\Gamma(7)$ denote the congruence subgroup of the modular group PSL(2, Z), that means

${\begin{pmatrix} a & b \\ c & d \end{pmatrix}}$

where a, b, c, d are integers,

$ad-cb = 1$

$a,d \equiv 1 \mod 7$

$b,c \equiv 0 \mod 7$

My question now is, what are the generators of this group? (and how would you derive that, if possible!)

I unfortunately don't have a clue, and I can find it nowhere.

Any help would be greatly appreciated,

Willem

PS: this is not a homework question

There is a very beautiful algorithmic approach to finding generators of congruence subgroups of $PSL(2, \mathbf{Z})$ using "Farey symbols" (sequences of rational numbers specifying a fundamental domain of a special kind). There is a survey article by Chris Kurth and Ling Long describing the algorithm and the theory behind it; and recent versions of the Sage computer algebra system have an implementation of Kurth and Long's algorithm (mostly due to Hartmut Monien).
masiao@fermat:~$sage ┌────────────────────────────────────────────────────────────────────┐ │ Sage Version 6.1.1, Release Date: 2014-02-04 │ │ Type "notebook()" for the browser-based notebook interface. │ │ Type "help()" for help. │ └────────────────────────────────────────────────────────────────────┘ sage: Gamma(7).generators() [ [1 7] [-48 7] [-209 56] [113 -35] [-55 21] [120 -49] [0 1], [ -7 1], [ -56 15], [ 42 -13], [-21 8], [ 49 -20], [ 15 -7] [ 239 -140] [113 -70] [ 232 -161] [-181 133] [ 8 -7] [ 28 -13], [ 70 -41], [ 21 -13], [ 49 -34], [ -49 36], [ 7 -6], [-76 105] [ 169 -238] [ 43 -63] [ 309 -490] [ 134 -217] [-21 29], [ 49 -69], [ 28 -41], [ 70 -111], [ 21 -34], [ 281 -476] [-230 399] [ 15 -28] [-97 231] [ 218 -525] [ 49 -83], [ -49 85], [ 7 -13], [-21 50], [ 49 -118], [-279 763] [ 22 -63] [-118 399] [ 29 -112] [-139 609] [ -49 134], [ 7 -20], [ -21 71], [ 7 -27], [ -21 92], [ 36 -175] [ 43 -252] [ 7 -34], [ 7 -41] ]  That command took less than 0.02 seconds to run. The theory actually guarantees that this is a free generating set, so these generators give an isomorphism between$\Gamma(7)\$ and the free group on 29 generators.