Fix $N \in \mathbb{N}$. Over $\mathbb{C}$, the modular curve $X_0(N)$ is the smooth compact modular curve compactifying the coarse moduli curve $Y_0(N)$ of pairs $(E,C)$, where $E$ is an elliptic curve over $\mathbb{C}$ and $C \le E[N]$ is a cyclic subgroup of order $N$. The boundary $X_0(N) \setminus Y_0(N)$ is the finite set of cusps, analytically identified with $\Gamma_0(N)\backslash\mathbb{P}^1(\mathbb{Q})$; in the generalized elliptic curve compactification these cusps correspond to degenerate generalized elliptic curves carrying compatible $\Gamma_0(N)$-level structure.