Discriminant-index formula for an integral lattice
= Discriminant-index formula for an integral lattice
Let $\omega_1,\ldots,\omega_n$ 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 satisfy
$$
G_\alpha=A^TG_\omega A,
$$
so
$$
\Delta(\alpha_1,\ldots,\alpha_n)
=(\det A)^2d_K.
$$
The absolute discriminant is minimized exactly when $A$ is unimodular, equivalently when the $\alpha_i$ form an integral basis.