Suppose we have an elliptic curve $E$ over $K$, an $l$-adic field. Say that $|j(E)|>1$ where $|.|$ is the $l$-adic valuation. By the theory of the Tate curve $E$ is isomorphic over $L$ to a Tate curve $E\_q$, where $L/K$ is a field extension of $K$ of degree at most 2.


In the case $[L:K]=2$, then $E(L)\cong E\_q(L) \cong L^\*/q^{\mathbb{Z}}$.
I was wondering how $E(K)$ looks like. $E(K)\cong E\_q^d(K)$ where $E\_q^d(K)$ is a quadratic twist by $d$ of $E\_q$. If $E\_q(K)\cong K^\*/q^{\mathbb{Z}}$, does $E(K)$ also have a "nice" form like that?



Twist of the Tate Curve
Chris provided a reference, but for those who don't have a copy of the book:
$$
E(K) \cong \bigl\{ u\in L^\*/q^{\mathbb Z} : \operatorname{\textsf{Norm}\_{L/K}}(u) \in q^{\mathbb Z}/q^{2\mathbb Z} \bigr\}.
$$


