User BrauerManinobstruction's question "Weil–Châtelet group of a real elliptic curve is isomorphic to Z/2Z when Δ>0" relates to exercise 10.7 of Silverman's AEC, which asks the reader to prove that for $E/\mathbb{R}$ an elliptic curve:
$$WC(E/\mathbb{R})=\mathbb{Z}/n\mathbb{Z}$$
where $n=1$ if $\Delta<0$ and $n=2$ if $\Delta>0$.
I am wondering if this could be part of a broader phenomenon with elliptic curves over local fields.
By local field, I specifically mean characteristic 0. Let $K$ be a local field, $E$ an elliptic curve, and $L/K$ a finite Galois extension.
Is $H^1(L/K, E)$ always finite? Is it cyclic?
Note that $H^1(L/K, E)$ is the kernel of $H^1(K, E)\rightarrow H^1(L, E)$; in other words, the group of $K$-torsors under $E$ with an $L$-rational point.
My motivation is the analogy in Galois cohomology of $E$ with the group of units of the ring of integers of $K$. For $L$ and $K$ as above, we have the following exact sequence (using Theorem 90):
$$1\rightarrow\pi_K^\mathbb{Z}\rightarrow\pi_L^\mathbb{Z}\rightarrow H^1(L/K, \mathcal{O}_L^\times)\rightarrow 1$$
where $\pi_K$ and $\pi_L$ are the uniformizers. This realizes $H^1(L/K, \mathcal{O}_L^\times)$ as cyclic of order equal to the ramification of $L/K$.
This leads me to conjecture (in the hopeful sense, rather than the believing sense) that $H^1(L/K, E)$ is at least most often cyclic, and the order is related to the discriminants over $L$ and $K$. My guess is this could also have to do with reduction or the Neron model.
Unfortunately, I have too little knowledge of Galois cohomology or elliptic curves to solve this issue. I couldn't even figure out the exercise of AEC for the archimedean case. I get stuck trying to show that each element of $H^1(L/K, E)$ is killed by multiplying by a certain integer.
Could anybody shed some light? Relevant resources are also welcome.