The modular equation $\Phi_n(X,Y)$ is a polynomial in $\mathbf Z[X,Y]$ relating the modular invariant $j$ and the functions $j\left(\frac{a\tau+b}{c\tau+d}\right)$, where $ad-bc=n$.

For example, we have identically

$$\Phi_n(j(n\tau),j(\tau))=0.$$

It is often asserted that the coefficients of $\Phi_n(X,Y)$ are *astronomically* large even for small values of $n$. Why is this so? Is there some lower bound on the coefficients?