On trace-one generators of abelian cubic fields

Let $K$ be a tamely ramified abelian cubic number field with discriminant $D_K$. We prove that the number of trace-one monic integral polynomials with root field $K$ and height $H$ is equal to the number of ideals in the quadratic field $\mathbb{Q}(\sqrt{-3})$ with norm $H^2{D}_K^{-1\mathrm{/}2}$.