Computation of determinant of a lattice

Question

Define the following lattice in $\mathbb{Z}^2,$ $$G:=\{\mathbf{x} \in \mathbb{Z}^2:\exists \lambda \in \mathbb{Z} \ \text{such that} \ \mathbb{x}\equiv \lambda (1,1) \mod 5\}. $$ What is the determinant $g$ of this lattice ?

Answer 1

A possible basis is $(1,1),(0,5)$. To show that this is a base, it is enough to show that they span $G$. For $(x,y)\in G$, we see that $(x,y)=x(1,1)+\frac{y-x}{5}(0,5)$. The volume of the parallelotope is 5.