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> Let $l$ be a prime $\geq 5$. Does there exist a pair $E,E'$ of elliptic curves, both defined over the same number field $K$, which are not $l$-isogenous over $K$, but are $l$-isogenous over a *quadratic* extension?
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I feel that the answer is **yes**, though I cannot come up with an example. 


I would also like to ask what happens if in addition one assumes that $E$ has CM. I am hoping that the answer in this case is **no**. In this case both curves will have the same CM field.



Elliptic Curves isogenous only over an extension?
Take an elliptic curve $E$ (say, over the rationals) with CM and $\ell$ a prime that splits in the CM field. Now take $E'$ to be a twist of $E$, so $E, E'$ are not isomorphic over the rationals and not $\ell$ isogenous either, as such an isogeny, composed with a self-isogeny will force them to be isomorphic. Now, over the field where they become isomorphic, they are also $\ell$ isogenous.


