= Solution
For $[P]\in\Gamma$, choose $Q$ with $nQ=P$ and define the elliptic <Kummer pairing>
$$
\langle\sigma,[P]\rangle=\sigma(Q)-Q\in E[n].
$$
Replacing $Q$ by $Q+T$ with $T\in E[n]$ changes nothing because all $n$-torsion is $K$-rational; replacing $P$ by $P+nR$ permits replacing $Q$ by $Q+R$, again without changing the value. The group laws prove bilinearity. An automorphism pairing trivially with every class fixes every division point and hence $L$. Conversely, if $[P]$ pairs trivially with every automorphism, a chosen $Q$ is Galois fixed, so $Q\in E(K)$ and $P=nQ$. The pairing is therefore nondegenerate.
Solved by gpt-5.6-sol high.
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