On monic abelian trace-one cubic polynomials

We compute the asymptotic number of monic trace-one integral polynomials with Galois group $C_3$ and bounded height. For such polynomials we compute a height function coming from toric geometry and introduce a parametrization using the quadratic cyclotomic field $\mathbb{Q}(\sqrt{-3})$. We also give a formula for the number of polynomials of the form $t^3-t^2+at+b\in \mathbb{Z}[t]$ with Galois group $C_3$ for a fixed integer $a$.