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In this paper we compute the degree of a curve which is the image of amapping $z\longmapsto (f(z): g(z): h(z))$ constructed out of three linearlyindependent modular forms of the same even weight $\ge 4$ into $\mathbb P^2$.We prove that in most cases this map is a birational equivalence and definedover $\mathbb Z$. We use this to construct models of $X_0(N)$, $N\ge 2$, usingmodular forms in $M_{12}(\Gamma_0(N))$ with integral $q$--expansion. The modelshave degree equal to $\psi(N)$ (a classical Dedekind psi function). When genusis at least one, we show the existence of models constructed using cuspidalforms in $S_4(\Gamma_0(N))$ of degree $\le \psi(N)/3$ and in$S_6(\Gamma_0(11))$ of degree 4. As an example of a different kind, we computethe formula for the total degree i.e., the degree considered as a polynomial oftwo (independent) variables of the classical modular polynomial (or the degreeof the canonical model of $X_0(N)$).

Integral Models of $X_0(N)$ and Their Degrees