Intersection of modular polynomials

In this paper, we consider the intersection of classic modular polynomials. The intersection number on affine space is given by the well-known Hurwitz class numbers. We give two different ways to compute the intersection number by two different compactifications of $\mathbb{A}^2$. This yields a new and more elementary formula for the intersection number. Consequently we get a class number relation. We also give a pure combinatorial proof of this class number relation.