Are there any congruence subgroups other than $\operatorname{SL}\_2(\mathbb Z)$ which have exactly 1 cusp? By congruence subgroup, I mean a subgroup of $\operatorname{SL}\_2(\mathbb Z)$ containing $\Gamma(N)$ for some $N$.


This question was motivated by the comments on the answer to [Generators of the graded ring of modular forms](https://mathoverflow.net/questions/66819/generators-of-the-graded-ring-of-modular-forms/211351#211351). In particular, if there are no such subgroups, then all congruence subgroups other than $\operatorname{SL}\_2(\mathbb Z)$ are generated in degree at most 5, with relations in degree at most 10, as follows from the comments on the answer to [Generators of the graded ring of modular forms](https://mathoverflow.net/questions/66819/generators-of-the-graded-ring-of-modular-forms/211351#211351).


I have a feeling the answer to this question may be well known, but I could not find it in Diamond and Shurman or by a Google search.



Are there congruence subgroups other than $\operatorname{SL}_2(\mathbb Z)$ with exactly 1 cusp?
Yes, there are plenty of others with the best known being the one corresponding to the hexagonal torus. For a list of low-genus congruence subgroups see [here.](http://www.uncg.edu/mat/faculty/pauli/congruence/) (Sebastian Pauli at UNC Greensboro).


