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?



Why are the coefficients of the modular equation so large?
Paula Cohen, [On the coefficients of the transformation polynomials for the elliptic modular
function,](https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/on-the-coefficients-of-the-transformation-polynomials-for-the-elliptic-modular-function/6D1315068F3E83B031CBBBCE853AAD9C) Math. Proc. of the Cambridge Phil. Soc. **95**, 389–402 (1984).


The largest coefficient $c\_{m}$ in absolute value of $\Phi\_{m}$ is of order
$$c\_{m}\simeq m^{km},$$
with $k$ a number of order unity ($k=9$ for $m=2^n$). So for $m=2^{10}=1024$ one has $c\_m\simeq 10^{27743}$ --- astronomically large, indeed.


