Coclass of the second 3-class group

TL;DR

This paper establishes a clear relationship between the coclass of second 3-class groups of number fields and the structure of their unramified cyclic cubic extensions, supported by extensive computational data.

Contribution

It proves that the coclass of the second 3-class group is uniquely determined by the second largest 3-class group among unramified extensions, using parametrized presentations.

Findings

Coclass is determined by the second largest 3-class group.

Minimal discriminants are computed for fields with given coclass.

Extensive database analysis supports the theoretical results.

Abstract

By means of parametrized presentations of finite metabelian 3-groups, it is proved that the coclass cc(M) of the second 3-class group M=Gal(F_3^2(K)/K) of any algebraic number field K with elementary bicyclic 3-class group Cl_3(K)=(3,3) is determined unambiguously by the second largest order ord(Cl_3(E_2))=3^{cc(M)+1} among the four 3-class groups of the unramified cyclic cubic extensions E_i (i=1,..,4) of K. Minimal discriminants of quadratic and cubic fields K with assigned coclass cc(M) are computed from extensive databases of 3-class numbers ord(Cl_3(E_i)) as an application.