Collinear CM-points

André’s celebrated Theorem of 1998 implies that each complex straight line Ax + By + C = 0 (apart from obvious exceptions) contains at most finitely many points $(j(\tau),j(\tau^\prime))$, where $\tau, \tau^\prime \in \mathbb{H}$ are algebraic of degree 2. We show that there are only a finite number of such lines which contain more than two such points. As there is a line through any two complex points, this is best possible.