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For an integral lattice, $L$, with Gram matrix $G$, the determinant of the lattice with respect to $G$ is the determinant of the Gram matrix, denoted $dG$.

Suppose $G$ and $G'$ are two distinct Gram matrix representations of a lattice $L$, then $dG'=dG\cdot \epsilon^2$, for $\epsilon\in \Z$. Accordingly, we define the discriminant ideal of a lattice by $dL=dG\cdot \Z^\times/{\Z^\times}^2$, which is independent of choice of basis.