  A result of the second-named author states that there are only finitely many
CM-elliptic curves over $\mathbb{C}$ whose $j$-invariant is an algebraic unit.
His proof depends on Duke's Equidistribution Theorem and is hence
non-effective. In this article, we give a completely effective proof of this
result. To be precise, we show that every singular modulus that is an algebraic
unit is associated with a CM-elliptic curve whose endomorphism ring has
discriminant less than $10^{15}$. Through further refinements and
computer-assisted computations, we eventually rule out all remaining cases,
showing that no singular modulus is an algebraic unit. This allows us to
exhibit classes of subvarieties in $\mathbb{C}^n$ not containing any special
points.
