I'm having some issues with abelian varieties and fields of definition. This already became clear in my previous question on Jacobians. Here's another question. If somebody can explain some nice facts on fields on definition this would help me a lot (because these aren't the only questions I have concerning fields of definition).
Let $A$ be an abelian variety over a number field $K$. Let $L/K$ be a finite field extension. Suppose that there exists an isogeny $A_L\to B$ defined over $L$.
Is this isogeny defined over $K$ if $B$ can be defined over $K$?
I think the answer is negative, but I have to admit that my set of examples is very scarce and, therefore, I can't find an easy counterexample.