Suppose we are given a map $f: X \to Y$ between two Riemann Surfaces, with branch points $p_1,p_2,\dots,p_n$ and known multiplicities at these points. Assuming we have a basis of $H_1(Y, \mathbb{Z})$, is there a standard choice of generators for $H_1(X, \mathbb{Z})$ in terms of the information about the branched points and the given basis?

The special case I have in mind is the spectral curve of some integrable system, and it is a double cover of a fixed elliptic curve with two branch points.

Are there good references for this?