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

Florian Breuer, Desirée Gijón Gómez, Fabien Pazuki

We give explicit upper and lower bounds on the size of the coefficients of the modular polynomials ${\Phi }_N$ for the elliptic $j$-function. These bounds make explicit the best previous…