Suppose $K$ is a number field and $E$ is an elliptic curve defined over $K$. *My question is:* how do you compute the local cohomology group $H^1(K\_v, \, E[p^{\infty}])$?


As to why I'm asking this, it comes up in [Iwasawa theory for elliptic curves](https://arxiv.org/abs/math/9809206) by Greenberg in page 74. In these notes, Greenberg is looking at the elliptic curve $E = 11a3$ defined over $\mathbf{Q}$. He claims that
$$H^1(\mathbf{Q}\_{11}, E[5^{\infty}]) \cong \mathbf{Z}/5\mathbf{Z}. $$
I'd like to understand how one would prove this result. Greenberg's notes say that this follows from the local duality theorems, but I'm not exactly sure how it follows from them. In terms of what I tried to figure this out, I tried to use the fact that $E[5^{\infty}]$ sits in an exact sequence of $G\_{\mathbf{Q}\_{11}}$-modules:
$$0 \to \mu\_{5^{\infty}} \to E[5^{\infty}] \to \mathbf{Q}\_5/\mathbf{Z}\_5 \to 0.$$
I tried applying the long-exact sequence in cohomology to this sequence, but I still couldn't show how to derive the result above. Does anyone know how to derive the fact above that $H^1(\mathbf{Q}\_{11}, E[5^{\infty}]) \cong \mathbf{Z}/5\mathbf{Z}$? Is there some other fact that I need to complete the calculation?



Calculating the Galois cohomology group $H^1(K_v, \, E[p^{\infty}])$
The corollary 3.4 of chapter 1 Arithmetic Duality theorems by Milne says that:
If $K$ has char 0 local field, there is a canonical pairing $$H^r(K,A^t)\times H^{1-r}(K,A)\to \mathbb{Q}/{\mathbb{Z}}$$ so because eliptic curves are self-dual for computing $H^1(K,E)$ it is enough to compute $E(K)$, It also implies $H^2(K,E)=0$.
Now just look at the exact sequence $$0\to E(5^n)\to E\to E\to 0$$.


