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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.