I am looking for a proof of the following result.
Let $\{ e_i\}_{i = 1}^n \in \mathbb{Z}^n \subset \mathbb{R}^n$ be linearly independent vectors, $E = \mathbb{Z} \langle e_1, \ldots e_n \rangle$ and $A \cong \mathbb{Z}^n/E$. Denote by $g(.,.)$ the standard inner product in $\mathbb{R}^n$. Then $|A| = \sqrt{detg(e_i,e_j)}$ where $|A| $ is the size of $A$.
I have figured it out. Denote by $R$ the matrix of the relations, i.e. the coordinates of $e_i$ in the standard basis of $\mathbb{R}^n$.
Note that by elementary transformations it can be made diagonal and its determinant is equal to the size of $A$ (structure of modules over euclidean rings). Note also that elementary transformations do not change the determinant.
The determinant of $g(e_i,e_j) $ is equal to $detRR^{T}$ so the result follows.