Let $E/\Bbb{Q}$ be an elliptic curve. $H^1(G_{\Bbb{Q}},E[2])$ is bijection with the set of pair of torsors and divisor of degree $2$. Let call the latter set $WCD(E/\Bbb{Q})$.
I tried to construct a bijection between them.
Is the following correct ? I'm having difficulty showing its well-definedness, so I my try is not on the right track.
Take $[C,[D]]\in WCD(E/\Bbb{Q})$ where $C$ is a curve and $[D]$ is a divisor class. Let fix an $\overline{\Bbb{Q}}$-isomorphism $f: E\cong C$ such that $[n]\circ f$ is defined over $\Bbb{Q}$. Then $\sigma \mapsto \sigma(f)-f$ where $\sigma(f)$ is action where $\sigma$ acts on its coefficients.
Then, $n\circ (\sigma(f)-f)=0$ and we can define cockle from $[C,[D]]$.
But how can we take $f$ such that $f: E\cong C$ such that $[n]\circ f$ is defined over $\Bbb{Q}$? Or am I go on a wrong track, how can I correctly define a cocycle from torsor divisor pair $[C,[D]]$ ?