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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?

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Paula Cohen, On the coefficients of the transformation polynomials for the elliptic modular function, 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.

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