A singular modulus is a $j$-invariant of a CM elliptic curve. It is known that it is always an algebraic integer. In 2015 Habegger proved that at most finitely many singular moduli are algebraic units. It was a special case of his more general ``Siegel Theorem for Singular Moduli''. Unfortunately, this result was not effective, because Siegel's zero was involved (through Duke's equidistribution theorem).

In the present work we obtain an explicit bound: if $\Delta=Df^2$ is an imaginary quadratic discriminant such that the corresponding singular moduli are units, then $|\Delta|<10^{75}$ and $|D|<10^{25}$.

Joint work with Philipp Habegger and Lars Kühne.