# of Karim Belabas, Aïssa Derhem, and Daniel C. Mayer

 * On these pages, we present most recent results of our joint research, directly from the lab. * Basic bibliography: K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 1213-1237 A. Derhem, Capitulation dans les extensions quadratiques non ramifiées de corps de nombres cubiques cycliques, Thèse de doctorat, Université Laval, Québec, 1988 D. C. Mayer, Multiplicities of dihedral discriminants, Math. Comp. 58 (1992), 831-847 and S55-S58 * Web master's e-mail address: contact@algebra.at *

 Inverse Population of Principal Factorization Types in certain classes of Pure Cubic Fields (2002/10/22) Dan (02/10/22): Inspired by studying the recent paper  by Moulay Chrif Ismaïli and Rachid El Mesaoudi, I investigated two interesting classes of conductors f in my recent most extensive database  of the 827600 pure cubic fields Q( R1/3 ) with R < 106. In these two classes of conductors, all three Principal Factorization Types ALPHA, BETA, and GAMMA can occur. Recall that the Overall Population of the Principal Factorization Types was: PFT BETA : 635434 fields or 76.78 % (absolutely ambiguous principal ideals exist), PFT ALPHA : 163530 fields or 19.76 % (relatively ambiguous principal ideals exist), PFT GAMMA : 28636 fields or 3.46 % (some unit in the normal field has norm zeta = exp(2 pi i/3) = [-1+(-3)1/2]/2). However, for the following two classes of conductors we have just the Inverse Statistical Population.

The first class of conductors f and radicands R is:
(*) f = 9p, R = 3p,9p with p = 1 (mod 9).

f R e u v m condition SK SN type p.f. # examples
9p 3p,9p 2 1 1 2 p = 1 (mod 9) 6523
(3/p)3=1 (3x,3y),x > 0,y > 0 ALPHA 754 R = 813,1569,2757
(3/p)3=1 (3x,3y),x > 0,y > 0 BETA 3,9 1202 R = 219,921,1731,1839
(3/p)3!=1 (3) (3,3) GAMMA 4374 R = 57,111,171,327
(3/p)3=1 (3x,3y),x > 1,y > 0 GAMMA 193 R = 657,2763,5193

Thus, we observe an Inverse Population of the Principal Factorization Types
among these 6523 fields:
PFT GAMMA: 4567 fields or 70.01 %,
PFT BETA: 1202 fields or 18.43 %,
PFT ALPHA: 754 fields or 11.56 %.

The second class of conductors f and radicands R is:
(**) f = pq, R = pq,p2q,pq2 with p = 1 (mod 9), q = 8 (mod 9).

f R e u v m condition SK SN type p.f. # examples
pq pq,p2q,pq2 0 2 0 2 p = 1,q = 8 (mod 9) 5948
(q/p)3=1 (3x,3y),x > 1,y > 0 ALPHA 741 R = 1241,2771,3401
(q/p)3=1 (3x,3y),x > 0,y > 0 BETA p,q 1081 R = 1853,2033,3383
(q/p)3!=1 (3) (3,3) GAMMA 4003 R = 323,629,1007,1349
(q/p)3=1 (3x,3y),x > 1,y > 0 GAMMA 123 R = 26333,44783,48149

Again, we observe an Inverse Population of the Principal Factorization Types:
among these 5948 fields:
PFT GAMMA: 4126 fields or 69.37 %,
PFT BETA: 1081 fields or 18.17 %,
PFT ALPHA: 741 fields or 12.46 %.

 References:  Moulay Chrif Ismaïli and Rachid El Mesaoudi, Sur la divisibilité exacte par 3 du nombre de classes de certain corps cubiques purs, Ann. Sci. Math. Québec 25 (2001), no. 2, 153 - 177  Daniel C. Mayer, Principal Factorization Types of Multiplets of Pure Cubic Fields Q( R1/3 ) with R < 106, Univ. Graz, Computer Centre, 2002