Let $E/k$ be an elliptic curve over some algebraically closed field $k$ of characteristic $p\ge 0$. It's known that $Aut(E)$ acts faithfully on the Tate module $T\_\ell(E)$ ($\ell\ne p$) with determinant 1. Is there a complete description of the actions of $Aut(E)$ on $T\_\ell(E)$? Ie, for any elliptic curve as above, can we describe the subgroup $Aut(E)\subset SL\_2(\mathbb{Z}\_\ell)\subset Aut(T\_\ell(E))$ up to conjugacy?


In characteristic 0 the action can be computed analytically and we find that the "extra automorphisms" $i,\rho$ of orders 4,6 (corresponding to $j$-invariant 1728,0) essentially act via conjugates of


$$M\_i =\begin{bmatrix}0 & 1\\-1 & 0\end{bmatrix}\qquad M\_\rho = \begin{bmatrix}1 & 1 \\ -1 & 0\end{bmatrix}$$


Reducing mod $p$ one finds that the reductions $\overline{i},\overline{\rho}$ act on the $\ell$-power torsion in the same way, and so the matrices are the same.


Thus, my question reduces to: If $char(k) = 2$ or 3, and $j = 0\equiv 1728$, then we have automorphism groups of order either 12 (characteristic 3), or 24 (characteristic 2). 


In this case what is the subgroup $Aut(E)\subset SL\_2(\mathbb{Z}\_\ell)$? We certainly get both the automorphisms $\overline{i},\overline{\rho}$ with matrices $M\_i,M\_\rho$, but there isn't necessarily a *single* choice of basis for $T\_\ell(E)$ such that $\overline{i},\overline{\rho}$ have matrices $M\_i,M\_\rho$ respectively. Also, in characteristic 2, there are additional automorphisms which aren't in the group generated by $\overline{i},\overline{\rho}$.


I feel like this must have been done somewhere, but I can't find any references for this.



how do automorphisms of elliptic curves act on the Tate module?
How many subgroups of order $12$ and $24$ are there in $SL\_2(\mathbb Z\_l)$? By the classification of finite subgroups of $SO(3)$ there are just two of each, one abelian and one non-abelian. It is easy to see that the abelian one cannot appear because the characteristic polynomial of each element should be integral. The non-abelian ones of order $12$ and $24$ are the degree $2$ central extensions of $S\_3$ and $A\_4$ respectively.


A more abstract computation would be observing that the endomorphism algebra is a quaternion algebra ramified at $p$ and $\infty$ and computing the group of units of this algebra. 


