= Covolume of a fractional ideal lattice
{title2=$\sqrt{|D_K|}\,N(\mathfrak b)$}
Let $j(\mathfrak b)$ be the <Minkowski embedding of a number field> applied to a nonzero <fractional ideal>. Under the metric $|x|^2=\sum_{v\text{ real}}x_v^2+2\sum_{v\text{ complex}}|x_v|^2$, its <covolume> is $\sqrt{|D_K|}N(\mathfrak b)$. For ordinary coordinate <Lebesgue measure> $\prod dx_v\prod d\operatorname{Re}z_v\,d\operatorname{Im}z_v$, its <covolume> is instead $2^{-r_2}\sqrt{|D_K|}N(\mathfrak b)$. Here $D_K$ is the <field discriminant>, and the absolute <norm of a fractional ideal> is positive and multiplicative. The formula follows by taking the embedding determinant of an <integral basis>, then using the index of an <integral ideal> and scaling to handle a <fractional ideal>.
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