Let $X$ be an abelian surface over $\mathbb{C}$. And let $i:X\longrightarrow X$ be the inverse map. $i$ is a degree 2 morphism. We consider $Y$ the quotient of $X$ by the action of $i$, that is, $Y=X/\langle i\rangle$. Now this is a singular surface whose singularities are 16 nodes. $\tilde{Y}$, the desingularization of $Y$ is a K3 surface.
My question is as follows. We have two canonical maps. $\pi: X\longrightarrow Y$ and the blow up map $f:\tilde{Y}\longrightarrow Y$. What is the push forward of the canonical line bundle $K$ of $X$ under $\pi$. I am guessing that it is trivial since $ K $ is trivial (since $X$ is an abelian surface) and the canonical bundle of $\tilde{Y}$ (which is isomorphic to $Y$ outside the exceptional locus) is trivial. Is this right? Thanks in advance!