 Level-two modular curve

The level-two modular curve is the quotient $Y(2)=\Gamma (2)\backslash \mathbb{H}$, where $\Gamma (2)$ is the level-two principal congruence subgroup. The effective group $\Gamma (2)/\{\pm I\}$ is torsion-free, so there are no elliptic stabilizers in its action on the complex upper half-plane. The compactification $X(2)$ is a genus-zero Riemann surface with three added modular cusps.

The modular lambda function identifies $Y(2)$ biholomorphically with $\mathbb{C}\setminus \{0,1\}$, and extends to an identification of $X(2)$ with the Riemann sphere. The three modular cusps map to $0,1,\infty$. Reduction modulo two gives projective subgroup index six, and each cusp width is two. The absence of elliptic stabilizers is a property of this group action, rather than a claim that the Riemann surface has no automorphisms.