# 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 * E-mail addresses: Karim.Belabas@math.u-psud.fr aderhem@yahoo.fr danielmayer@algebra.at *

 Pure Cubic Fields with 3-class rank 1, 2, or 3 (2002/10/20) Dan (02/10/20): The possible conductors f and radicands R in Ismaïli's 3 configurations of pure cubic fields K = Q( R1/3 ) with 3 || hK and SN = (3,3) are summarized in the following 3 tables. (Again, I use the concepts and symbols of the first article in this series.) The multiplicity of the conductor m = m(f), i. e., the number of non-isomorphic fields sharing f and forming a so called multiplet of pure cubics , is a function of u, v, and e: u = #{ q prime | q divides f, q congruent 1,8 (mod 9) } (free prime divisors), v = #{ q prime | q divides f, q incongruent 1,8 (mod 9) } (restrictive prime divisors), and e = v3(f) is the exact 3-exponent of f that can be used to characterize Dedekind's types 1A (e = 2), 1B (e = 1), and 2 (e = 0). We have m(f) = 2u * Xv-1, where Xi = [2i-(-1)i]/3. Not all of these fields occur as singulets (m = 1). Many of them come as doublets (m = 2) or quartets (m = 4). In pseudo-singulets, the companion radicand exeeds 106 in  . If the Principal Factorization Type is BETA and if there is a unique orbit of principal factors, two principal factors (p.f.) are listed, which are minimal, in general (except when p < p2 < q or q < q2 < p). To permit a feeling how frequently these radicands occur I give the counts (#) in my recent most extensive table  of the 827600 pure cubic fields Q( R1/3 ) with R < 106 and all occurrences below 100 (or higher, if necessary) as examples.

1st configuration:

The fields with e = 2, u = 0, v = 2
arise in 3514 complete doublets (m = 2) and 6020 pseudo singulets
( 2*3514 + 6020 = 13048 = 8709 + 4339 );
2348 doublets are of type (ALPHA,ALPHA), 1166 of type (BETA,BETA),
but no inhomogeneous family types (ALPHA,BETA) occur;
4013 pseudo singulets are of type (ALPHA), 2007 of type (BETA).

f R e u v m condition SK SN type p.f. # examples
p p 0 1 0 1 p = 1 (mod 9) 13063
(3w),w > 0 (3w,3w-1) ALPHA 11958 R = 19,37,73
(3w),w > 1 (3w,3w) GAMMA 1105 R = 541,919,1279
3p p 1 0 2 1 p = 4,7 (mod 9) 26168
(3/p)3!=1 (3) (3) ALPHA 17485 R = 7,13,31,43,79,97
(3/p)3=1 (3) (3,3) BETA 3,9 8683 R = 61,67,103,151
9p 3p,9p 2 0 2 2 p = 4,7 (mod 9) 13048
(3/p)3!=1 (3) (3) ALPHA 8709 R = 21,39,63,93
(3/p)3=1 (3) (3,3) BETA 3,9 4339 R = 183,201,309,453
pq pq,p2q,pq2 0 0 2 1 p = 4,7,q = 2,5 (mod 9) 29615
(q/p)3!=1 (3) (3) ALPHA 19898 R = 26,28,35
(q/p)3=1 (3) (3,3) BETA p,q 9717 R = 62,172,287

We see that the
possible pure cubic fields K = Q( R1/3 ) of Ismaïli's 1st configuration
constitute 81894 of 827600, i.e., about 9.90% of all fields
in the range R < 106 of normalized radicands R.

2nd configuration:

The fields with e = 2, u = 1, v = 1
arise in 1758 complete doublets (m = 2) and 3022 pseudo singulets
( 2*1758 + 3022 = 6538 = 4835 + 1703 );
800 doublets are of type (BETA,BETA), 17 of type (GAMMA,GAMMA),
and here also 941 inhomogeneous family types (BETA,GAMMA) occur;
2294 pseudo singulets are of type (BETA), 728 of type (GAMMA).

f R e u v m condition SK SN type p.f. # examples
9q 3q,9q 2 1 1 2 q = 8 (mod 9) 6538
(3w),w > 0 (3w,3w) BETA 3,9 4835 R = 51,159,213
(3w),w > 0 (3w,3w) GAMMA 1703 R = 153,321,477
q1q2 q1q2,q12q2 0 2 0 2 q1,q2 = 8 (mod 9) 3007
(3w),w > 0 (3w,3w) BETA q1,q2 2259 R = 901,1819,3043
(3w),w > 1 (3w,3w) GAMMA 748 R = 1207,1513,3763
3q1q2 q1q2,q12q2 1 0 3 1 q1,q2 = 2,5 (mod 9) (3) (3,3) BETA 21460 R = 20,22,58,92,94
3q1q2 q1q2,q12q2 1 1 2 2 q1 = 2,5 (mod 9),q2 = 8 (mod 9) (3w),w > 0 (3w,3w) BETA 27510 R = 34,68,85,106
9q1q2 3xq1yq2z,x,y,z = 1,2 2 0 3 4 q1,q2 = 2,5 (mod 9) (3w),w > 0 (3w,3w) BETA 20999 R = 30,60,66,90
9q1q2 3xq1yq2z,x,y,z = 1,2 2 1 2 4 q1 = 2,5 (mod 9),q2 = 8 (mod 9) (3w),w > 0 (3w,3w) BETA 13171 R = 102,204,255,306
q1q2q3 q1xq2yq3z,x,y,z = 1,2 0 0 3 1 q1,q2,q3 = 2,5 (mod 9) (3) (3,3) BETA 5249 R = 460,550,638,820
q1q2q3 q1xq2yq3z,x,y,z = 1,2 0 1 2 2 q1,q2 = 2,5 (mod 9),q3 = 8 (mod 9) (3w),w > 0 (3w,3w) BETA 9661 R = 170,530,710,748

Therefore, the
possible pure cubic fields K = Q( R1/3 ) of Ismaïli's 2nd configuration
constitute 107595 of 827600, i.e., about 13.00% of all fields
in the range R < 106 of normalized radicands R.

3rd configuration:

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
(3x,3y),x > 0,y >= 0 GAMMA 4567 R = 57,111,171,327
pq pq,p2q,pq2 0 2 0 2 p = 1,q = 8 (mod 9) 5948
(q/p)3=1 (3x,3y),x > 0,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
(3x,3y),x > 0,y >= 0 GAMMA 4126 R = 323,629,1007,1349
3pq pq,p2q,pq2 1 0 3 1 p = 4,7,q = 2,5 (mod 9) 29714
(q/p)3=1 (3x,3y),x > 0,y > 0 ALPHA 7147 R = 65,86,124,203
(3x,3y),x > 0,y >= 0 BETA 22567 R = 14,52,77,122
3pq pq,p2q,pq2 1 1 2 2 p = 4,7,q = 8 (mod 9) 14286
(q/p)3=1 (3x,3y),x > 0,y > 0 ALPHA 880 R = 2759,3521,3893
(3x,3y),x > 0,y >= 0 BETA 13406 R = 119,221,371,497
3pq pq,p2q,pq2 1 1 2 2 p = 1,q = 2,5 (mod 9) 27285
(q/p)3=1 (3x,3y),x > 0,y > 0 ALPHA 3303 R = 209,254,407,614
(3x,3y),x > 0,y >= 0 BETA 23982 R = 38,74,76,95,146
9pq 3xpyqz,x,y,z = 1,2 2 0 3 4 p = 4,7,q = 2,5 (mod 9) 28491
(3v,3w),v > 0,w >= 0 ALPHA 4137 R = 126,231,468,474
(3v,3w),v > 0,w >= 0 BETA 24354 R = 42,78,84,105
9pq 3xpyqz,x,y,z = 1,2 2 1 2 4 p = 4,7,q = 8 (mod 9) 6478
(3v,3w),v > 0,w >= 0 ALPHA 414 R = 5271,6201,8277
(3v,3w),v > 0,w >= 0 BETA 6064 R = 357,663,1071
9pq 3xpyqz,x,y,z = 1,2 2 1 2 4 p = 1,q = 2,5 (mod 9) 13002
(3v,3w),v > 0,w >= 0 ALPHA 1515 R = 342, 1086,1332
(3v,3w),v > 0,w >= 0 BETA 11487 R = 114,222,228,285
pq1q2 pxq1yq2z,x,y,z = 1,2 0 0 3 1 p = 4,7,q1,q2 = 2,5 (mod 9) 10813
(3v,3w),v > 0,w >= 0 ALPHA 2632 R = 658,1925,2060
(3v,3w),v > 0,w >= 0 BETA 8181 R = 154,260,350,406
pq1q2 pxq1yq2z,x,y,z = 1,2 0 1 2 2 p = 4,7,q1 = 2,5 ,q2 = 8 (mod 9) 10292
(3v,3w),v > 0,w >= 0 ALPHA 580 R = 1378,2485,5516
(3v,3w),v > 0,w >= 0 BETA 9712 R = 442,476,595,1054
pq1q2 pxq1yq2z,x,y,z = 1,2 0 1 2 2 p = 1,q1,q2 = 2,5 (mod 9) 9455
(3v,3w),v > 0,w >= 0 ALPHA 1054 R = 730,1270,2755
(3v,3w),v > 0,w >= 0 BETA 8401 R = 190,370,836,874

We see that the
possible pure cubic fields K = Q( R1/3 ) of Ismaïli's 3rd configuration
constitute 162287 of 827600, i.e., about 19.61% of all fields
in the range R < 106 of normalized radicands R.

In contrast to the previous tables,
the pure cubic fields with the following conductors have already 3-rank rK = 2 or 3:

f R e u v m condition SK type p.f. # examples
p1p2 p1p2,p12p2 0 0 2 1 p1,p2 = 4,7 (mod 9) 8210
(3x,3y),x,y > 0 ALPHA 7731 R = 91,217,469
(p1/p2)3=(p2/p1)3=1 (3x,3y),x,y > 0 BETA p1,p2 479 R = 2359,3241,5383
3p1p2 p1p2,p12p2 1 0 3 1 p1,p2 = 4,7 (mod 9) 8170
(3x,3y,3z),x,y > 0,z >= 0 ALPHA 6040 R = 301,403,427,553
(3x,3y,3z),x,y > 0,z >= 0 BETA 2130 R = 1519,1687,1939,2509

 References:  Daniel C. Mayer, Principal Factorization Types of Multiplets of Pure Cubic Fields Q( R1/3 ) with R < 106, Univ. Graz, Computer Centre, 2002