The domain: a Riemann surface realized as two copies of the complex plane branched at the points
$Zm\kappa_2:=-\frac{1}{\kappa_2}$, $Zm\kappa_1:=-\frac{1}{\kappa_1}$, $Zm1:=-1$,
$Z1:=1$, $Z\kappa_1:=\frac{1}{\kappa_1}$, $Z\kappa_2:=\frac{1}{\kappa_2}$
and compactified with two points at infinity, where the light green section meets the blue section, and the green section meets the light blue section as shown below.
The gray lines represent segments of the surface where the branches switch between the top and bottom plane.
Equivalently: a hyperelliptic plane curve of genus two realized as the zero locus
$$ \{\, (x,y)\in\mathbb{C}^2 \,|\, y^2 = (1-x^2)(1-\kappa_1^2x^2)(1-\kappa_2^2x^2) \,\} .$$
The Abel-Jacobi embedding of this surface into its Jacobian is given by
$$ (x,y) \;\longrightarrow\; \left( \int_{(0,1)}^{(x,y)} \omega_1 \,,\, \int_{(0,1)}^{(x,y)} \omega_2 \right) \;\mathrm{mod}\, \Pi\cdot\mathbb{Z}^4 \,, $$
where $\omega_1=\frac{dx}{y},\,\omega_2=\frac{x\,dx}{y}$ are the hyperelliptic differentials of first and second kind, and $\displaystyle \Pi$ is the period matrix $(\int_{\alpha_j}\omega_i)_{ij}$ with respect to the cycles
$ \alpha_1: Zm1 \xrightarrow{+} Z1 \xrightarrow{-} Zm1 $
,
$ \alpha_2: Zm\kappa_2 \xrightarrow{+} Zm\kappa_1 \xrightarrow{-} Zm\kappa_2 $
,
$ \alpha_3: Zm1 \xrightarrow{-} Zm\kappa_1 \xrightarrow{+} Zm1 $
,
$ \alpha_4: Zm\kappa_2 \xrightarrow{-} Z\kappa_2 \xrightarrow{+} Zm\kappa_2 $
.
Removing the cycles $\alpha_1,\alpha_2,\alpha_3,\alpha_4$ from the domain, one obtains a simply connected Riemann surface which is mapped biholomorphically onto its image in the Jacobian. This image is visualized above when all periodicity values are set to 1.
The image of the simply connected Riemann surface under one single differential $\omega_1$ or $\omega_2$ is visualized by setting the projection of the above visualization to $\mathbb{C}^2\to\mathbb{C}\times\{0\}$ or $\mathbb{C}^2\to\{0\}\times\mathbb{C}$ repsectively. In the first case, one will recognize the familiar doubly periodic rectangular pattern in the complex plane traced by the elliptic differential of the first kind, as described in [Stein&Shakarchi, Section 9.4.5].
