I know there are some similar questions to this, but I have a theorem for clasification of covering spaces:
Given a topological space $B$ (path conncted and locally path connected).
$\phi: Cub(B) \to Conj(\pi_1(B,b))$ such as $[p:E \to B] \to [p_{\ast}(\pi_1(E,e))]$
where $Cub(B)$ is the set of class of isomorphism of covering spaces over $B$, and $Conj(\pi_1(B,b))$ is the set of classes of conjugation of subgroups of $\pi_1(B,b)$.
So $\pi_1(S^1 \times S^1)=\mathbb{Z} \times \mathbb{Z}$ and its abelian. So the $Conj(\mathbb{Z} \times \mathbb{Z})$ are the subgroups of $\mathbb{Z} \times \mathbb{Z}$.
but since they are asking the 3-sheeted covering.
I have found that there are two possibilities:
$A=<(3,0),(0,1)> =\{(3m,n), m,n \in \mathbb{Z}\}$ and $B=<(3,1),(0,1)>$
such as $r_1: \mathbb{R}^2/A \cong S^1 \times S^1 \to \mathbb{R}^2/\mathbb{Z}^2 \cong S^1 \times S^1$ and for $r_2: \mathbb{R}^2/B \cong S^1 \times S^1 \to \mathbb{R}^2/\mathbb{Z}^2 \cong S^1 \times S^1$.
is it possible for $C=<(0,1),(0,3)>$ ?
Am I missing another covering space of 3-sheetes of $S^1 \times S^1 $.
Thanks in advance