Let \(\ell\geq 2\) be an integer and consider a cubic equation \(f_\ell(X)= X^3+\ell X-1=0\). This equation has a unique position real root \(v < 1\). The \(k_\ell:=\mathbb{Q}(v)\) is a real cubic field with a unique imaginary place. Let \(O_\ell\) denote the ring of integers of \(k_\ell\). If \(4\ell^3+ 27\) is squarefree then \(O_\ell\) coincides with \(\mathbb{Z}[v]\). The authors show that under these conditions there is a unique unary perfect form over \(k_\ell\) up to homothety and \(\mathrm{GL}_1(O_\ell)\)-equivalence.