  Let $E$ be an elliptic curve and $\pi:E\to\mathbb{P}^{1}$ a standard double
cover identifying $\pm P\in E$. It is known that for some torsion points
$P_{i}\in E$, $1\leq i\leq4$, the cross ratio of $\{\pi(P_{i})\}_{i=1}^{4}$ is
independent of $E$. In this article, we will give a complete classification of
such quadruples.
