A359310 Cyclic cubic conductors associated with closed Andozhskii groups.

59031, 209853, 247437, 263017, 271737, 329841, 377923, 407851, 412909, 415597, 416241, 416727, 462573, 474561, 487921, 493839, 547353, 586963, 612747, 613711, 615663, 622063, 648427, 651829, 689347, 690631, 753787, 796779, 811069, 818217, 869611, 914263, 915439, 922167, 936747, 977409, 997087

Offset: 1

Author

Daniel Constantin Mayer, Dec 25 2022

Keywords

• nonn
• changed

Comments

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]

Examples

			Cyclic cubic number fields with conductors 59031, respectively 209853, respectively 247437, 263017, 271737, elementary tricyclic 3-class group (3,3,3), and harmonically balanced capitulation have been discovered by Daniel Constantin Mayer on 13 July 2022, respectively 15 July 2022, respectively 25 December 2022. Each of them belongs to a quartet of non-isomorphic fields sharing a common conductor, such that the other three fields have 3-class group (3,3) and capitulation type (1243), called G.16. The conductors bigger than 300000 were computed by Bill Allombert at the University of Bordeaux with PARI/GP.
		

Links

• Bill Allombert, Plateforme Fédérative pour la Recherche en Informatique et Mathématique (PlaFRIM)
• Bill Allombert and Daniel Constantin Mayer, Cyclic cubic number fields with harmonically balanced capitulation, arXiv:2307.13898 [math.NT], 2023.
• Bill Allombert and Daniel Constantin Mayer, Corps de nombres cubiques cycliques ayant une capitulation harmonieusement équilibrée, Publications mathématiques de Besançon. Algèbre et théorie des nombres (2025), pp. 21-46. See p. 23.
• I. V. Andozhskii and V. M. Tsvetkov, On a series of finite closed p-groups, Izv. Akad. Nauk SSSR, Ser. Mat. 38 (1974), no. 2, 278-290.
• I. V. Andozhskii, On some classes of closed pro-p-groups, Izv. Akad. Nauk SSSR, Ser. Mat. 39 (1975), no. 4, 707-738.
• Daniel Constantin Mayer, Theoretical and experimental approach to p-class field towers of cyclic cubic number fields, Four plenary lectures and exercises, Les Sixièmes Journées d'Algèbre, Théorie des Nombres et leurs Applications (JATNA), 25-26 November 2022, Oujda, Morocco, pp. 83-86.