Let $N \in \mathbb{N}$. The *modular curve* $X_0(N)$ is an algebraic curve defined over $\Q$ whose points correspond to pairs $(E,\phi)$, where $E$ is an elliptic curve and $\phi$ is an isogeny with domain $E$ and kernel a cyclic group of order $N$, with the exception of particular points called *cusps* which correspond to degenerate such pairs.

The set of its complex points $X_0(N)(\C)$ is naturally isomorphic to the quotient of the completed upper half plane $\Gamma_0(N) \backslash \mathcal{H}^*$ as a Riemann surface.

**Authors:**

**Knowl status:**

- Review status: beta
- Last edited by Nicolas Mascot on 2015-12-14 18:24:31.874000