What is the $\Gamma(1)$?

Question

$\Gamma(N)=\Big\{ \begin{pmatrix} a &b\\ c& d \end{pmatrix}\in SL_{2}(\mathbb{Z}) | \begin{pmatrix} a &b\\ c& d \end{pmatrix} \equiv \begin{pmatrix} 1 &0\\ 0& 1 \end{pmatrix} \mod N\Big \}$.

What is the $\Gamma(1)$? Is it the $SL_{2}(\mathbb{Z})$?

Answer 1

Yep because$\pmod 1$, everything is equivalent.

Answer 2

Two integers are always equivalent modulo $1$. Indeed $a-b = 1 \times (a-b).$ So there is actually no condition in your set and $\Gamma(1) = SL_2(\mathbb{Z}).$