Are there any good way to understand $k$-polarized Abelian surfaces? I am aware that if $A \cong \mathbb{C}^2/\Gamma$ is $k$-polarized, the lattice $\Gamma$ can be taken of the form 
$$
\begin{bmatrix}
1 & 0 & \tau\_1 & \tau\_2\\
0 & k & \tau\_3 & \tau\_4
\end{bmatrix}
$$
over $\mathbb{Z}$ (think of $\mathbb{C}^2\cong \mathbb{R}^4$) such that the imaginary part of 
$
\begin{bmatrix}
\tau\_1 & \tau\_2\\
\tau\_3 & \tau\_4
\end{bmatrix}
$
is positive definite. Are there any other good way to see $k$-polarized Abelian surfaces?



Intuitive meaning of $k$-polarized Abelian surface?
It is well known that any polarized abelian variety is isogenous to a principally polarized one. More precisely, given any polaized abelian variety $(A, \, L)$ there exists a principally polarized abelian variety $(B,\, \Theta)$ and an isogeny $u \colon A \longrightarrow B$ such that $L=u^\* \Theta$.


Then any polarized abelian surface $(A, \, L)$, where $L$ is of type $(1, \,k)$, admits an isogeny of degree $k$ over a principally polarized abelian surface $(B, \, \Theta)$ which is compatible with the polarizations. In other words, $(A, \, L)$ is an étale cover (necessarily Galois, with abelian Galois group) of $(B, \, \Theta)$. 


For instance, $A$ admits a polarization of type $(1, \,2)$ if and only if it is an étale double cover of a principally polarized abelian surface.


