I would like to rephrase a question I asked three days ago and was closed for the unclear presentation.

Taking $T \in E[p]$, a point in the p-torsion of the elliptic curve and looking at the right action of $SL_2({\mathbb Z}/p{\mathbb Z})$ on $T$ for prime $p$, one can easily see that you obtain again all points in $E[p]$ equally many times. The action is described as follows:

$$ \Big(\frac{m}p \quad \tau+ \frac{n}p \Big) \left(\begin{matrix} a & b \\\ c & d \end{matrix} \right) = \Big(\frac{ma+nc}p \quad \tau+ \frac{mb+nd}p \Big) $$

Now what would happen when we take $T \in E[n]$ where $n$ is an integer and look at the right action of $SL_2({\mathbb Z}/p{\mathbb Z})$? Do we also get again all the points in $E[n]$ equally many times? If not, why?