Edit: $a,b,c,d$ are fixed in advance. Otherwise it is clearly $\mathbb{Z}^2$. Apologies.
Given $a,b,c,d \in \mathbb{Z}$ such that $ad-bc\neq 0$. Set $\Gamma:=\langle (a,b),(c,d)\rangle$, what is its index in $\mathbb{Z}^2$?
I know this group has finite index in $\mathbb{Z}^2$, as you can quickly see that $(ad-bc,0)$ and $(0,ad-bc)$ are elements of $\Gamma$, and the index of $\langle (ad-bc,0),(0,ad-bc)\rangle$ in $\mathbb{Z}^2$ is $(ad-bc)^2$.