The tori \(T\) and \(\mathcal G_1 = R^E_\mathbb {Q}\mathbb {G}_m\) are isomorphic as algebraic groups over \(\mathbb {Q}\). Every rational point \((K/\mathbb {Q},x)\) of \(T\) thereby determines an element of \(\mathbb {Q}(\sqrt{-3})^\times \) which is canonically determined up to the action of \(\mathrm{Aut}(\mathcal G_1)\). For integral rational points of \(T\), the toric height \(H_{\mathrm{tor}}(f)\mathrel {:=}\sqrt{1-3a}\) is identified with the square-root of the norm on \(\mathbb {Q}(\sqrt{-3})^\times \). Let \(\zeta \) be a primitive cube root of unity. If \(u+v\zeta \in \mathbb {Q}(\sqrt{-3})^\times \) has norm \(N\) and trace \(T\), then the characteristic polynomial of the corresponding rational point \((K/\mathbb {Q},x)\) is

Such a polynomial either has Galois group \(C_3\) or splits into three linear factors over \(\mathbb {Q}\), with at most two linear factors being the same. Conversely, a monic trace-one polynomial \(f=t^3-t^2+at+b \in \mathbb {Q}[t]\) which either has Galois group \(C_3\) or splits into three linear factors over \(\mathbb {Q}\), with at most two linear factors being the same, can be expressed in this way for precisely two rational points of \(T\) if \(f\) has no repeated roots, or for precisely one rational point of \(T\) if \(f\) has a double root which is not a triple root. The elements \(u+v\zeta \in \mathbb {Q}(\sqrt{-3})^\times \) corresponding to \(f=t^3-t^2+at+b\) will be the roots of the quadratic polynomial

The polynomial \(f\) will have integral coefficients if and only if