Let $p=2$ or 3, and let $k$ be an algebraically closed field of char. $p.$ Let $E$ be the supersingular elliptic curve over $k$ (with $j=0$). Let $G$ be the automorphism group of $E,$ which has order 12 (resp. 24) when $p=3$ (resp. 2). Then the $\ell$-adic cohomology $H^1(E,\mathbb Q\_{\ell})$ is a 2-dimensional representation of $G.$ Is it irreducible?


Since we know the group structure of $G$ (cf. Silverman's AEC, Appendix A, exercise..), does anyone have a reference for its character table?



supersingular elliptic curve in char. 2 or 3
If you have a supersingular curve in general, the $\ell$-adic cohomology is a faithful representation of the endomorphism ring, which is a quaternion order that splits over $\mathbb{Q}\_\ell$. In the case of characteristic 2 and 3, the automorphism groups are non-abelian (in particular, isomorphic to $SL\_2(\mathbb{F}\_3)$ and $\mathbb{Z}/3\mathbb{Z} \rtimes \mathbb{Z}/4\mathbb{Z}$, respectively). For non-abelian groups, any faithful two dimensional representation (away from characteristic dividing the order) is automatically irreducible. You can find character tables of the groups in GAP, using commands like:



```
g := SpecialLinearGroup(2,3);
tbl := CharacterTable(g);
Display(tbl);

```

In SAGE, you can add "gap" to the beginnning of things:



```
g = gap.SpecialLinearGroup(2,3)
tbl = gap.CharacterTable(g)
gap.Display(tbl)

```

There might be a better way to do this.


