Associate family of Scherk's First and Fifth Surfaces

$\varphi=\pi/4$

Weierstrass Data

\[ M=(\mathbb{C}\cup\{\infty\})\setminus\{e^{i\varphi},e^{-i\varphi},-e^{i\varphi},-e^{-i\varphi}\}, \] \[ g=z,\qquad \eta = \frac{e^{i\theta} dz}{z^4-2\cos (2\varphi)z^2+1}, \] where $\theta,\,\varphi\in\mathbb{R}$ are constants.