$\newcommand{\C}{{\mathbb{C}}}\newcommand{\Q}{{\mathbb{Q}}}\renewcommand{\Im}{{\mathrm{Im}}}$ Let $\tau$ be an imaginary quadratic number with ${\Im\tau>0}$ and let $j$ denote the $j$-invariant function. According to the classical theory of Complex Multiplication, the complex number $j(\tau)$ is an algebraic integer. A *CM-point* in $\C^2$ is a point of the form $(j(\tau_1),j(\tau_2))$, where both $\tau_1$ and $\tau_2$ are imaginary quadratic numbers.