Let $E/K$ be an elliptic curve over a field $K$. Then, how $\sigma \in \text{Gal}(L/K)$ acts on first Galois cohomology group $H^1(G_L,E) $ where $G_L$ denotes absolute Galois group of $L$?
$H^1(G_L,E)\cong WC(E/K)$, where $WC(E/K)$ is a group of torsos of $E/K$. For torsor, $\text{Gal}(L/K)$ acts on its coefficients, that is, the action is given by $C\to C^{\sigma}$. What is the corresponding action to $H^1(G_L,E)$ ?
The functor $-^{G_L}\colon G_L\text{-mod}\to\mathrm{Ab}$ factors through as $$G_L\text{-mod}\xrightarrow{\bullet^{G_L}} \mathrm{Gal}(L/K)\text{-mod}\xrightarrow{\text{forget}}\mathrm{Ab}.$$ Since the forgetful functor $\mathrm{Gal}(L/K)\text{-mod}\to\mathrm{Ab}$ is exact, the derived functors of the composition $\bullet^{G_L}\colon G_L\text{-mod}\to\mathrm{Ab}$, which is exactly the Galois cohomology $H^i(G_K,\bullet)$ is the composition of the $i$-th derived functor of $G_L\text{-mod}\xrightarrow{\bullet^{G_L}} \mathrm{Gal}(L/K)\text{-mod}$ and the forgetful functor. In other words, $H^i(G_K,\bullet)$ has a natural $\mathrm{Gal}(L/K)$-module structure.