Let $E$ be an elliptic curve over a field $K$. I am mostly interested in the case $K$ is a number field or a local field but the following question is valid for all $K$.

Let $p$ be any prime power which is comprime to the characteristics of $K$. Galois cohomology gives a cup product map

$$H^{1}(K,E[p]) \times H^{1}(K,E[p]) \to H^{2}(K,E[p] \otimes E[p])$$

Composing this map with the map induced by the Weil pairing gives a map

$$ \phi: H^{1}(K,E[p]) \times H^{1}(K,E[p]) \to H^2(K,\mu_{p}). $$

Are there any known examples where the above map $\phi$ is non-trivial?

In theory, it is possible to compute both sides explicitly when p is small. However, computing the cup product is quite hard (for me.)