Let $K\subset L$ be a finite field extension.
Let $X$ and $Y$ be (smooth projective geometrically connected) curves over $L$.
Let $f:X\to Y$ be a finite morphism of curves over $L$.
Assume that $Y$ can not be defined over $K$.
Is it possible that $X$ can still be defined over $K$?
Equivalently, suppose that $X$ can be defined over $K$. Then is it true that $Y$ can be defined over $K$?
So basically, I'm asking about the following. If you have a curve $X$ over some field $K$ and a finite morphism $X_L\to Y$ over some extension $L/K$, can we define the curve $Y$ over $K$?
I suspect the answer to be no. But I can't seem to find an easy counterexample.
I was thinking about taking $Y$ to be an elliptic curve which can't be defined over $\mathbf{Q}$ (irrational $j$-invariant) and constructing a suitable branched cover...