Singularities of the Modular Curve

Let X0(ℓ) be the modular curve, parameterizing cyclic isogenies of degree ℓ, and Z0(ℓ) be its plane model, given by the classical modular equation Φℓ(X, Y)=0. We prove that all singularities of Z0(ℓ), except two cusps, are intersections of smooth branches, and evaluate the order of contact of these branches.