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 ΦN for any N ≥ 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.