A result of Habegger shows that there are only finitely many singular moduli such that $j$ or $j-alpha$ is an algebraic unit. The result uses Dukes Equidistribution Theorem and is thus not effective. For a fixed $j$-invariant $alpha in bar{mathbb{Q}}$ of an elliptic curve without complex multiplication, we prove that there are only finitely many singular moduli $j$ such that $j-alpha$ is an algebraic unit. The difference to the work of Habegger is that we give explicit bounds.
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