Regulator of a number field

Consider the \(\mathop{\mathrm{\mathbf{log}}}\) embedding defined on the article of DUT. According to the DUT, we know that \(\log(\mathcal{O}_{K}^{*}) \subseteq (1,1,1,\cdots)^{\perp} \subseteq \mathbb{R}^{r_1+r_2}\) is a discrete subgroup, cocompact in \((1,1,1\cdots)^{\perp}\).

Consider the map \(\pi: \mathbb{R}^{r_1+r_2} \rightarrow \mathbb{R}^{r_1+r_2-1}\), which is defined to be just the projection on the first \(r_1+r_2-1\) coordinates (effectively forgetting the last coordinate). Then the set \(\pi \circ \mathop{\mathrm{\mathbf{log}}}(\mathcal{O}_{K}^{*})\) is a lattice inside \(\mathbb{R}^{r_1+r_2 -1}\). The covolume is defined to be the regulator of the field \(K\) and is denoted by \(R_{K}\).

Computationally

The above definition agrees with the output of the regulator as defined on sage.

# WARNING: Works pretty much only for rank of number field <10 

KQ = 7
Kfield = CyclotomicField(KQ)
print Kfield.regulator()