For positive integers n and L, denote by SL_n(Z,L) the level L congruence subgroup of SL_n(Z), ie the kernel of the homomorphism SL_n(Z)-->SL_n(Z/LZ).

For n at least 3, it is known that SL_n(Z,L) is normally generated (as a subgroup of SL_n(Z)) by Lth powers of elementary matrices. Indeed, this is essentially equivalent to the congruence subgroup problem for SL_n(Z).

However, this fails for SL_2(Z,L) since SL_2(Z) does not have the congruence subgroup property.

Question : Is there a nice generating set for SL_2(Z,L)? I'm sure this is in the literature somewhere, but I have not been able to find it.

For positive integers $n$ and $L$, denote by $SL_n(Z,L)$ the level $L$ congruence subgroup of $SL_n(Z)$, i.e. the kernel of the homomorphism $SL_n(Z)\rightarrow SL_n(Z/LZ)$.

For $n$ at least $3$, it is known that $SL_n(Z,L)$ is normally generated (as a subgroup of $SL_n(Z)$) by Lth powers of elementary matrices. Indeed, this is essentially equivalent to the congruence subgroup problem for $SL_n(Z)$.

However, this fails for $SL_2(Z,L)$ since $SL_2(Z)$ does not have the congruence subgroup property.

Question : Is there a nice generating set for $SL_2(Z,L)\ ?$ I'm sure this is in the literature somewhere, but I have not been able to find it.