One can view this surface as a translation invariant version of the Finite Riemann surface.
The quotient under the horizontal translation has two annular and two catenoidal ends, all with vertical limiting normal. The López-Ros factor changes the growth rates of the catenoidal ends.
Other placements of the catenoidal necks are possible:
Their locations are controlled by elementary versions of Traizet’s balance equation. In a configuration with one bottom catenoidal neck and n top catenoidal necks of equal neck size, arranged in the line, these locations are determined by the roots of Legendre polynomials. The details have been worked out by Peter Connor.
Resources
P. Connor: A Note on Special Polynomials and Minimal Surfaces
Mathematica Notebooks for one top end and two top ends
PoVRay Sources for one top end and two top ends
Minimal Surfaces 