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?
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.