Let $j(\tau)$ be the modular invariant. Let $\mathcal M_n^*$ be the set of all $2$-by-$2$ matrices with relatively prime integer entries. The modular equation is defined by $$\Phi_n(X,j(\tau))=\prod_{M\in \operatorname{SL}_2(\mathbf Z)\backslash\mathcal M_n^*}(X-j(M\tau)).$$ What is the Galois group of the polynomial $\Phi_n(X,X)$?

Let $j(\tau)$ be the modular invariant. Let $\mathcal M_n^*$ be the set of all $2$-by-$2$ matrices with relatively prime integer entries and with determinant $n$. The modular equation is defined by $$\Phi_n(X,j(\tau))=\prod_{M\in \operatorname{SL}_2(\mathbf Z)\backslash\mathcal M_n^*}(X-j(M\tau)).$$ What is the Galois group of the polynomial $\Phi_n(X,X)$?

For more details on the modular equation, see

Zagier: Elliptic Modular Forms and Their Applications, p. 68

Andrew Sutherland: The modular equation