$\mod 2$ Galois representation of an elliptic curve

Question

I was given only simple example how to explicitely find $\mod 2$ Galois representation of an elliptic curve (i.e. the first example on page 2 HERE) but how can I find matrices in $GL_2(F_2)$ for other cubic polynomials, for example$f(x)=x^3+x+3$? Since the discriminant of this polynomial is not a square I know that Galois group of its extension field is isomorphic to symmetric group $S_3=\left< \sigma,\tau| \sigma^2=\tau^3=id, \sigma \tau \sigma=\tau^{-1} \right>$, but I'm not sure how to use this to find matrices for this polynomial.

Answer 1

If your elliptic curve is given by affine equation $y^2=x^3+ax+b$ then $2$-torsion points are precisely $0,(\alpha_1,0),(\alpha_2,0),(\alpha_3,0)$ where $\alpha_i$ are roots of $x^3+ax+b=0$. So, to compute the mod 2 Galois representaion is equivalent to compute the action of Galois group on these roots.

In your example let us choose $(\alpha_1,0),(\alpha_2,0)$ as a basis in $E[2]$, $(\alpha_3,0)=(\alpha_1,0)+(\alpha_2,0)$ so the matrices are $$\sigma=(12)\mapsto\left(\begin{matrix}0 & 1\\1 & 0\end{matrix}\right),\tau=(123)\mapsto \left(\begin{matrix}0 & 1\\1 & 1\end{matrix}\right) $$ and this is the mod 2 reduction of the 2-dimensonal irreducible representation of $S_3$.