  We obtain an explicit upper bound on the size of the coefficients of the
elliptic modular polynomials $\Phi_N$ for any $N\geq1$. 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.
