Explicit bounds on the coefficients of the modular polynomials and the size of $X_0(N)$

We give explicit upper and lower bounds on the size of the coefficients of the modular polynomials ${\mathrm{\Phi }}_N$ for the elliptic $j$-function. These bounds make explicit the best previously known asymptotic bounds. We then give an explicit version of Silverman’s Hecke points estimates. Finally, we give an asymptotic comparison between the Faltings height of the modular curve $X_0(N)$ and the height of the modular polynomial ${\mathrm{\Phi }}_N$.