Holomorphic map from the complex plane to a one-dimensional complex torus

Every holomorphic map $f:\mathbb{C}\to \mathbb{C}/\Lambda$ is constant or surjective. It lifts to an entire function $F:\mathbb{C}\to \mathbb{C}$. If $F$ is nonconstant, the Little Picard theorem says it omits at most one point, while every fiber of $\mathbb{C}\to \mathbb{C}/\Lambda$ is an infinite lattice coset; hence every point of the torus has a preimage under $f$.