Mod-three Galois representation of an elliptic curve (${\rho }_{E,3}:G_K\to {\mathrm{GL}}_2({\mathbb{F}}_3)$)

The absolute Galois group acts on $E[3]\simeq {\mathbb{F}}_3^2$, giving the mod-three Galois representation. Its invariant vectors are precisely the rational points of order dividing three.