Explicit bounds on the coefficients of modular polynomials for the elliptic $j$-invariant

We obtain an explicit upper bound on the size of the coefficients of the elliptic modular polynomials ${\mathrm{\Phi }}_N$ for any $N\ge 1$. These polynomials vanish at pairs of $j$-invariants of elliptic curves linked by cyclic isogenies of degree $N$. The main term in the bound is asymptotically optimal as $N$ tends to infinity.