This is probably a silly question but I'll ask it anyway. Here goes:
Let $\Gamma$ be a congruence subgroup of SL$_2$($\mathbb{Z}$). To each point $\tau \in \mathcal{H}$ (where $\mathcal{H}$ is the complex upper half plane) is associated the positive integer
$$h_{\tau} = |\{ \pm I \} \Gamma_{\tau} / \{ \pm I \} | = \begin{cases} |\Gamma_{\tau} | /2 & \text{if } -I \in \Gamma_{\tau}, \\ |\Gamma_{\tau}| & \text{if } -I \notin \Gamma_{\tau}. \end{cases}$$
$\Gamma_{\tau}$ is the isotropy subgroup for each elliptic point $\tau$ of $\Gamma$.
Why is it that $|\{ \pm I \} \Gamma_{\tau} / \{ \pm I \} | = |\Gamma_{\tau}|/2$ if $-I \in \Gamma_{\tau}$? Why is it suddenly halved?
Similarly, suppose $\Gamma_1 \subset \Gamma_2$ are two congruence subgroups of SL$_2$($\mathbb{Z}$) and let $f: X(\Gamma_1) \to X(\Gamma_2)$ be the natural project of corresponding modular curves. Why is (again only unsure of the first case)
$$[ \{\pm I\} \Gamma_2 : \{\pm I \}\Gamma_1] = \begin{cases} [\Gamma_2 : \Gamma_1]/2 & \text{if } -I \in \Gamma_2 \text{ and } -I \notin \Gamma_1,\\ [\Gamma_2 : \Gamma_1] & \text{otherwise}. \end{cases} $$ ? The explanations must be similar!
I hope you can help.
This mysterious formula for $h_\tau$ has a simple algebraic/geometric interpretation. The point is that the action of $\text{SL}_2 \mathbb{Z}$ on $\cal H$ is not faithful, its kernel is $\{\pm I\}$, and the action factors through a faithful action by the quotient group $\text{PSL}_2 \mathbb{Z} = \text{SL}_2 \mathbb{Z} / \{\pm I\}$.
Letting $P\Gamma_\tau$ denote the isotropy group of $\tau$ for the action of $\text{PSL}_2 \mathbb{Z}$ on $\cal H$, the surjective homomorphism $\text{SL}_2 \mathbb{Z} \to \text{PSL}_2 \mathbb{Z}$ restricts to a surjective homomorphism $\Gamma_\tau \to P\Gamma_\tau$ the kernel of which is $\{\pm I\}$ if $-I \in \Gamma_\tau$ and is trivial if $-I \not\in \Gamma_\tau$. Thus $$h_\tau = |P\Gamma_\tau| $$