Pushforward of structure sheaf of a curve under a finite map

Question

Let $f:X\rightarrow C$ be a double cover of smooth projective curves. We know that $f_*O_X$ is a rank two vector bundle and it splits as a direct sum of line bundles $O_C\oplus L$. Is it possible that $L=O_C$ as well. That is, can the push forward of structure sheaf be trivial?