X0 2 modular curve ($X_0(2)$)

The compact quotient for the Gamma 0 congruence subgroup at level two has projective index three, two modular cusps of cusp widths one and two, one order-two elliptic orbit and no order-three elliptic orbit. It has genus zero. It differs from the level-two principal-congruence curve for $\Gamma (2)$, which has three modular cusps.

The ratio of modular discriminants has weight zero and level ${\Gamma }_0(2)$. It has one simple pole at infinity and one simple zero at the zero modular cusp, with no zeros or poles in the half-plane. It is a spherical coordinate by the single-pole criterion for a spherical coordinate. The Klein j-invariant is $(t+256{)}^3/t^2$, with $t(e^{2\pi i/3})=-256$.