On the ${\text{PGL}}_2$-invariant quadruples of torsion points of elliptic curves

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 $P_i\in E$, $1\le i\le 4$, 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.