It is shown that a necessary condition for the existence of a bicolored Steiner triple system of order $n$ is that $n$ can be written in the form $A^2+3B^2$ for integers $A$ and $B$ . In the case when $n=q$ is either a prime congruent to 1 mod 3, or the square of a prime congruent to 2 mod 3, it is shown that the numbers of colored vertices in the triple system would be unique, and are given by the number of points on specific twists of the CM elliptic curve $y^2=x^3-1$ over the finite field ${\bb F}_q$ .