Let $N \in \mathbb{N}$, and let

\begin{align*} \Gamma_0(N) := \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbb{Z}) : c \equiv 0 \pmod N \right\}. \end{align*}

If $Y_0(N)(\mathbb{C})$ is identified analytically with $\Gamma_0(N)\backslash \mathbb{H}$, where $\mathbb{H} := \{z \in \mathbb{C} : \operatorname{Im}(z) > 0\}$, then there is a natural bijection between $Y_0(N)(\mathbb{C})$ and the set of isomorphism classes of pairs $(E,C)$, where $E$ is a complex elliptic curve and $C \le E[N]$ is a cyclic subgroup of order $N$. Here an isomorphism $(E,C) \cong (E',C')$ is an isomorphism of complex elliptic curves $\varphi: E \to E'$ such that $\varphi(C)=C'$.