An effective "Theorem of Andre" for CM-points on a plane curve

It is a well known result of Y. Andre (a basic special case of the Andre-Oort conjecture) that an irreducible algebraic plane curve containing infinitely many points whose coordinates are CM-invariants is either a horizontal or vertical line, or a modular curve Y-0(n). Andre's proof was partially ineffective, due to the use of (Siegel's) class-number estimates. Here we observe that his arguments may be modified to yield an effective proof. For example, with the diagonal line X-1 + X-2 = 1 or the hyperbola X1X2 = 1 it may be shown quite quickly that there are no imaginary quadratic tau(1), tau(2) with j(tau(1))+ j(tau(2)) = 1 or j (tau(1)) j (tau(2)) = 1, where j is the classical modular function. 2010 MSC codes 11G30, 11G15, 11G18.