cp's OEIS Frontend

The coclass cc(M) for one of the fields K with conductor c = a(n) is n-1, and for each field K with conductor c < a(n), the coclass cc(M) is less than n-1. Among the 3-groups M of coclass cc(M)=1, we distinguish the abelian 3-group A=(3,3) by formally putting cc(A)=0, in accordance with the FORMULA. This is a significant difference to quadratic fields, which are firstly uniquely determined by their discriminant, and secondly cannot have an abelian second 3-class group.

The coclass cc(M) for one of the fields K with conductor c = a(n) is n-1, and for each field K with conductor c < a(n), the coclass cc(M) is less than n-1. Among the 3-groups M of coclass cc(M)=1, we distinguish the abelian 3-group A=(3,3) by formally putting cc(A)=0, in accordance with the FORMULA. This is a significant difference to quadratic fields, which are firstly uniquely determined by their discriminant, and secondly cannot have an abelian second 3-class group.

The coclass cc(M) for the field K with discriminant d = -a(n) is 2*n, and for each field K with absolute discriminant |d| < a(n), the coclass cc(M) is less than 2*n.

The coclass cc(M) for the field K with discriminant d=a(n) is n, and for each field K with discriminant d < a(n), the coclass cc(M) is less than n.

The Magma program "RealCoClass.m" in the Links is independent of the data file "ipad_freq_real" by M. R. Bush. It computes the first five terms, 180527768 inclusively, in precisely 14 days of CPU time on an Intel Core i7 4790 quadcore processor with clock rate 4.0 GHz.

According to their differential principal factors (DPF), the normal closures of pure cubic number fields can be classified into three types ALPHA, BETA, GAMMA (see Aouissi et al., Period. Math. Hungar.). For each type, the generating radicals (cube roots) are DPF. For type BETA, absolute DPF exist additionally. Type BETA can be subdivided further into three subtypes (see Aouissi et al., Kyushu J. Math.). For each subtype, the units form an orbit of lattice minima in the maximal order of the pure cubic field. For subtype M2, resp. M1, resp. M0, two, resp. one, resp. no, further orbit(s) of lattice minima, consisting of non-unital DPF with principal factor norms, exist additionally. The exotic subtype M0 has the fatal drawback that the Voronoi algorithm, which recursively constructs the chain of lattice minima, fails to detect non-unital DPF, although they exist, and thus is unable to find the correct classification into type BETA. While the coarse types ALPHA, BETA, GAMMA can be distinguished by means of MAGMA or PARI/GP, no modern computer algebra system possesses the required routines to resolve the fine subtypes M2, M1, and M0.

According to their ambiguous principal ideals (API), the normal closures of pure cubic number fields can be classified into three types (see Aouissi et al.). For each type, the generating radicals (cube roots) are API. For type Alpha, relative API exist additionally. For type Beta, absolute API exist additionally. For type Gamma, only the radicals are API, but some unit of the normal closure has a primitive third root of unity as its cyclic cubic relative norm. The latter property is characteristic for type Gamma. If the radicand is a prime p == 1 (mod 9) then type Beta is excluded, and, statistically, type Alpha dominates by far. The present subsequence of radicands with type Gamma is very sparse. Ismaili and El Mesaoudi have proved an important application of this subsequence. Type Gamma enables more capitulation types of closely related normal closures in their unramified cyclic cubic extensions than type Alpha.

The maximal unramified pro-7-extension, that is, the Hilbert 7-class field tower, of these imaginary quadratic fields must have a Schur sigma-group as its Galois group. The tower has an unbounded number of stages at least equal to two, and may even be infinite.

The maximal unramified pro-5-extension, that is, the Hilbert 5-class field tower, of these imaginary quadratic fields must have a Schur sigma-group as its Galois group. The tower has an unbounded number of stages at least equal to two, and may even be infinite.

An algebraic number field with this capitulation type has a 7-class field tower of precise length 2 with Galois group isomorphic to the Schur sigma-group SmallGroup(16807,7). It is a solution to the problem posed by Olga Taussky-Todd in 1970.

An algebraic number field with elementary tricyclic 3-class group and harmonically balanced capitulation, that is, a permutation in the symmetric group on 13 letters, such that precisely 1, respectively 7, transfer kernels satisfy the Taussky condition A, has a 3-class field tower of exact length 2 with Galois group a closed Andozhskii group of order 6561 or a sibling of order 2187 or the parent of order 729. The identifier of the group in the SmallGroups database is 217700+i with i=10,11,12, respectively i=1,2,3, or 4660+j with j=9,10,11,12, respectively j=1,2,3,4, or 136, respectively 133. When precisely 4 transfer kernels satisfy the Taussky condition A, the 3-class field tower may have two or three stages. [Corrected by Daniel Constantin Mayer, Apr 01 2023]

Below the bound c < 10^6, only four cyclic cubic fields with conductors c = 689347, 753787, 796779, 869611 possess a closed Andozhskii group of order 6561 as 3-class field tower group. Ten with conductors c = 59031, 415597, 416727, 462573, 487921, 493839, 547353, 622063, 915439, 936747 have a non-closed group of order 2187. The remaining 23 conductors give rise to non-closed groups of order 729. [Supplemented by Daniel Constantin Mayer, Jul 21 2023]

Beyond c > 10^6, another closed Andozhskii group of order 6561 is realized by the conductor c = 1406551. [Supplemented by Daniel Constantin Mayer, Sep 24 2023]