Modular curves \(X_0(N)\) with infinitely many quartic points

We determine all modular curves \(X_0(N)\) $X_0(N)$ with infinitely many quartic points. To do this, we define a pairing that induces a quadratic form representing all possible degrees of a rational morphism from \(X_0(N)\) $X_0(N)$ to a positive rank elliptic curve.