I am looking the group of automorphisms $G$ of the curve defined over $\mathbb F\_3$ by (in projective coordinates) $Y^2Z=X(X-Z)(X-2Z)$.
Obviously, there are the automorphisms $X\mapsto X+\alpha Z$ (for $\alpha\in\mathbb F\_3$) and $Y\mapsto\pm Y$. But are they the only ones? 
And a second question: What is the field $(\mathbb F\_3(x)[y])^G$? I did not manage to determine it.



Automorphisms of a curve
This is the elliptic curve $E:y^2=x^3-x$ over $\mathbb F\_3$ whose $j$-invariant is $0$. The automorphism groups of elliptic curves over finite fields are quite well known, so this question is probably better suited for MathStackExchange. Over $\overline{\mathbb F}\_3$ the automorphism group has order 12; it is the semi-direct product of $C\_4$ and $C\_3$. The "other" automorphism, i.e., the one you need to compose with the ones that you already have, is $(x,y)\mapsto(-x,iy)$, where $i^2=-1$. but since $-1$ is not a quadratic residue mod 3, you need to go to the field $\mathbb{F}\_9$ to get all of $\operatorname{Aut}(E/
\overline{\mathbb F}\_3)$. 


