Discriminant-index formula for an integral lattice

Let ${\omega }_1,\dots ,{\omega }_n$ be an integral basis and let ${\alpha }_j={\sum }_i{a}_{ij}{\omega }_i$ be a rational basis of algebraic integers, with integer matrix $A=(a_{ij})$. Their trace Gram matrices satisfysoThe absolute discriminant is minimized exactly when $A$ is unimodular, equivalently when the ${\alpha }_i$ form an integral basis.