$\Gamma(2)$ and $\Gamma_0(4)$ are conjugate

Question

Show that the groups $\Gamma(2)$ and $\Gamma_0(4)$ are conjugate to each other in $SL(2,\mathbb{R})$, where $\Gamma(2)$ and $\Gamma_0(4)$ is congruence subgroups of the modular group

Is it the standard statement ?

Answer 1

Yes, this is true. Both groups have index $6$ in $PSL_2(\mathbb{Z})$, and they are conjugate by $\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$ in $PSL_2(\mathbb{R})$. For details see the classification of the conjugacy classes of the genus zero congruence subgroups of $PSL_2(\mathbb{R})$ with no elliptic elements by Sebbar here, Section $7$, in particualr page $388$.