Direct image of structure sheaf of a singular curve

Question

Suppose $f:X\rightarrow C$ is a degree two morphism of projective curves over complex numbers. Assume $C$ is smooth and $X$ has only nodal singularities at $d$ points and smooth elsewhere. So the images in $C$ of the $d$ points are the branch points, say $p_1,...p_d$. In this case what is $f_*O_X$? Is it $O_C\oplus O(-\Sigma\ p_i)$? Is there any restriction on $d$?