The regulator of $\mathbb Q$ and of imaginary quadratic fields is $1$

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Throughout, $K$ is a number field (a field of characteristic zero that is finite-dimensional over $\mathbb Q$), $\mathcal O_K$ is its ring of integers and $\mathcal O_K^\times$ the unit group. $r_1$ denotes the number of real infinite places of $K$ and $r_2$ the number of complex infinite places (each complex place corresponds to a pair of complex-conjugate embeddings $K\to\mathbb C$). $\operatorname{Reg}(K)$ is Mathlib's NumberField.Units.regulator K: the absolute value of the determinant of the $r\times r$ matrix $\big(N_w\log\lvert\varepsilon_i\rvert_w\big)$ where $\varepsilon_1,\dots,\varepsilon_r$ ($r=r_1+r_2-1$) is a fixed (Mathlib-chosen) system of fundamental units, i.e. units whose classes form a basis of $\mathcal O_K^\times$ modulo torsion, and $w$ ranges over the infinite places other than one fixed place; for $r=0$ it is the determinant of the empty matrix, $1$. Statement. Let $K$ be a number field such that $[K:\mathbb Q]=1$, or $[K:\mathbb Q]=2$ and $K$ is totally complex. Then $\operatorname{Reg}(K)=1$.