Let $K = \mathbb{Q}(\sqrt{ 2})$. Let $E_1$ be an elliptic curve over $K$. What is the structure of the set of points $E_1(K)$? Assume there is an elliptic curve $E_2$ defined over $K$ and a map $f : E_1 \rightarrow E_2$ defined over $K$ that is surjective with finite kernel. Compare $E_2(K)$ and $E_1(K)$ as precisely as possible.
My attempt:
From Mordell-Weil I know that $E_1(K) \cong \mathbb{Z}^r \bigoplus E_1(K)_{tors}$ and same for $E_2(K) \cong \mathbb{Z}^s \bigoplus E_2(K)_{tors}$. Can I say something more on the structure of $E_1(K)$??
Then we have that $f$ is surjective so my guess is that the rank of $E_2$ is less than the rank of $E_1$, i.e. $s \leq r$? I don't really know if this is right...
And the finite kernel implies that $f(E_1(K)_{tors}) \subseteq E_2(K)_{tors}$??
Any help?