Twist of the Tate Curve

Question

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?

Answer 1

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