Classify, except isomorphism, all the associated covering spaces of $3$ sheets on $\mathbb{S}^1\times\mathbb{S}^1$. Do the same for the covering spaces of $4$ sheets.
I know that $G=\mathbb{Z}^2$ acts by translation in $\mathbb{R}^2$ and $\pi:\mathbb{R}^2\to \mathbb{R}^2/\mathbb{Z}^2\cong\mathbb{S}^1\times\mathbb{S}^1$ is a universal covering. In this case $\pi_1(\mathbb{S}^1\times\mathbb{S}^1)=\mathbb{Z}^2$ and thus for each $A$ subgroup of $\mathbb{Z}^2$ there is a respective covering space $r:\mathbb{R}^2/A\to \mathbb{S}^1\times\mathbb{S}^1$. In the problem that I try to solve, I try to find all the subgroups of index $3$ and $4$ of $\mathbb{Z}^2$, what are these? The subgroups of $\mathbb{Z}^2$ have the form $n\mathbb{Z}\times m\mathbb{Z}$ where $n,m\in\mathbb{Z}$? I could take for example $A=\left \langle (3,0),(0,1) \right \rangle=\{(3m,n):m,n\in\mathbb{Z}\}$ and $B=\left \langle (3,1),(0,1) \right \rangle$, these are subgroups of $\mathbb{Z}^2$ with index $3$ such that $r_1:\mathbb{R}^2/A\cong \mathbb{S}^1\times\mathbb{S}^1\to \mathbb{R}^2/\mathbb{Z}^2\cong\mathbb{S}^1\times\mathbb{S}^1$ and $r_2:\mathbb{R}^2/B\cong \mathbb{S}^1\times\mathbb{S}^1\to \mathbb{R}^2/\mathbb{Z}^2\cong\mathbb{S}^1\times\mathbb{S}^1$ are covering spaces of $3$ sheets, is there more?