The quotient space of $\mathbb C$ obtained by identifying points differing by a Gaussian integer is topologically a torus. The map that takes each point in $\mathbb C$ to its corresponding point in the quotient space is a covering map. But the concept of "covering map" is, as far as I know, purely topological. These spaces have more structure than their topologies and in particular, this mapping is conformal. And I just might also want to view these as metric spaces with the usual Euclidean metric on this covering space and the shortest-path metric on the quotient space.
Is there some standard name for this particular covering map? If not, what name for it would be appropriate?