The question basically says it all - there is of course a natural action of $G_{\overline{K} / K}$ on $E[m]$, where $E$ is an elliptic curve defined over $K$ and $E[m]$ is the $m$-torsion subgroup of that curve. My question is whether this action is transitive.
This seems like a natural question, but I haven't been able to find anything that addresses the question directly.