Bound the difference of two singular moduli

For a fixed singular modulus $\alpha$, we give an effective lower bound of norm of $x-\alpha$ for another singular modulus $x$ with large discriminant. We then generalize this result for ${\mathrm{\Phi }}_m(x,\alpha )$, where $\Phi_m(X,Y) \in \Z[X,Y]$isthe$m$-th modular polynomial.