Let $X,Y$ be two elliptic curves over $K:=K(R)$ which have good models over $R$, Char($K$)=$0$, where $R$ complete DVR with algebraically residue field $k$.


If $L$ is a finite extension of $K$, such that $X \otimes L \cong Y \otimes L$.


My question is:


Is there $X \cong Y$ ?



morphism between two elliptic curves over a local field
I assume that by "good models" you mean models $\mathscr{X}$, $\mathscr{Y}$ which are elliptic $R$-curves. These are pointed curves of genus 1, and in particular stable curves. This implies that the sheaf 
$$I:=\underline{\rm Isom}\_{\text{$R$-ell. curves}}(\mathscr{X},\mathscr{Y})$$
is a scheme, finite and unramified over $R$. With your assumptions on $R$ (strictly henselian is enough), $I$ is then a finite disjoint sum of copies of closed subschemes of $R$. In particular, for every extension $L$ of $K$ (finite or not), we have $I(R)=I(L)$, which gives the result.


