Calculating the image of 4-torsion points under the map that maps 3-torsion points to roots of unity

Question

Given an elliptic curve $E/\mathbb{C}$, if I identify its 3-torsion points by involutions (i.e. project on to $\mathbb{P}^1$), and then map three of them to (1, $s$, $s^2$) using an element from PGL(2,$\mathbb{C}$), where $s$ is the third root of unity, do I know where the 4 torsion points (after identification by involution) will go to? Is there a way to describe them knowing something about $E$ (e.g. j-invariant, the equation, or the lattice parameter)?