By means of parametrized presentations of finite metabelian \(3\) -groups, it is proved that the coclass \(\textrm{cc}(M)\) of the second \(3\) -class group \(M=\textrm{Gal}(\textrm{F}_3^2(K)/K)\) of any algebraic number field \(K\) with elementary bicyclic \(3\) -class group \(\textrm{Cl}_3(K)\simeq (3,3)\) is determined unambiguously by the second largest order \(\textrm{ord}(\textrm{Cl}_3(E_2))=3^{\textrm{cc}(M)+1}\) among the four \(3\) -class groups of the unramified cyclic cubic extensions \(E_i\) \((i=1,\ldots ,4)\) of \(K\) . Minimal discriminants of quadratic and cubic fields \(K\) with assigned coclass \(\textrm{cc}(M)\) are computed from extensive databases of \(3\) -class numbers \(\textrm{ord}(\textrm{Cl}_3(E_i))\) as an application.
