# 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 de corps de nombres cubiques cycliques, Thèse de doctorat, Université Laval, Quebec, 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 *

*
 First occurrence of cubic discriminants with 3-defect 2 (2001/07/28)
*
Dan (01/07/28): Since 1997 I am very impressed by your paper
A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), no. 219, 1213-1237.

My own research concerns the multiplicities of cubic (and other) discriminants:
[1] Multiplicities of dihedral discriminants, Math. Comp. 58 (1992), no. 198, 831-847 and Supplements section S55-S58.
[2] Discriminants of metacyclic fields, Canad. Math. Bull. 36(1) (1993), 103-107.

Now I have extended the first of these two papers to more general situations,
which occur only for rather huge discriminants
that exceed the tables of my colleagues Hugh C. Williams and Jordi Quer by far.

So you are probably the unique insider who could give me some numerical confirmation for my extended theory.
I would be very glad, if you could verify the existence of the following complex cubic discriminants
and their multiplicities (non-isomorphic fields sharing the discriminant)
with the aid of your algorithm:

1. Symmetric configuration
4 discriminants with tau=3, delta_3(f)=2, (a1,...a4)=(1,1,1,0)

-212204300 (9 fields)
-350786700 (9 fields)
-687541932 (9 fields)
-4297137075 (9 fields)

2. Asymmetric configuration
1 discriminant with tau=2, delta_3(f)=1, (a1,...a4)=(2,0,0,0)
2 discriminants with tau=3, delta_3(f)=2, (a1,...a4)=(1,1,1,0)
1 discriminants with tau=4, delta_3(f)=2, (a1,...a4)=(2,1,1,0)

-5742252 (9 fields)
-86842700 (9 fields)
-1758564675 (9 fields)
-7034258700 (9 fields)

3. Multi layer configuration
1 discriminant with tau=3, delta_3(f)=2, (a1,...a4)=(1,1,1,0)
1 discriminants with tau=4, u=1, delta_3(f)=2, (a1,...a4)=(1,1,1,0)

-191575692 (9 fields)
-4789392300 (18 fields)

4. Asymmetric configuration
1 discriminant with tau=2, delta_3(f)=1, (a1,...a4)=(2,0,0,0)
2 discriminants with tau=3, delta_3(f)=2, (a1,...a4)=(1,1,1,0)
1 discriminants with tau=4, delta_3(f)=2, (a1,...a4)=(2,1,1,0)

-39468627 (9 fields)
-32618700 (9 fields)
-48726700 (9 fields)
-3946862700 (9 fields)

As an attachment I include some AMS-TEX pages (unfortunately in German) where you see how I came to these discriminants.
*
Karim: As far as magnitudes go, this should be relatively easy
[my current implementations achieve discriminant bounds about 10^14 in "decent" time,
esp. for single discriminants, as opposed to whole tables].
I'll check the multiplicities at the end of the holiday break (by mid-September),
"not overramified" real cubic fields [corresponding to unramified extensions of the quadratic subfield of their Galois closure].
Used them to prove that Quer's quadratic fields with 3-rank = 5 were indeed the smallest ones.
*
Karim: My programs are in perfect accordance with your expectations. Here are the fields.
The format is silly, it's just the way debugging output is spit out by the program: it goes as follows

disriminant: Hessian (A,B,C)=Ax^2+Bx+C Form [a,b,c,d]=ax^3+bx^2+cx+d

your (triples of isomorphic) fields being generated by any root of Form
[which is Mathews-Berwick-reduced, not Julia-reduced;
the latter might yield slightly smaller defining polynomials]

-212204300: (-13020,-19530,4900) [62,0,70,35]
-212204300: (-420,-28770,-113750) [1,15,215,3555]
-212204300: (4480,-4550,-34370) [3,67,1,171]
-212204300: (7000,-2730,-22470) [5,85,15,89]
-212204300: (-8610,-30030,-7700) [37,60,110,110]
-212204300: (-12740,-3990,12180) [38,16,114,17]
-212204300: (-5810,-24710,1120) [51,2,38,54]
-212204300: (-21350,-27930,-1680) [85,130,150,62]
-212204300: (-23240,-24710,280) [94,136,148,53]

-350786700: (630,-21510,-234000) [1,30,90,2690]
-350786700: (-1350,-38610,-81180) [3,27,231,1661]
-350786700: (900,-28890,-60480) [10,60,90,381]
-350786700: (-3960,-36630,-18270) [11,33,153,421]
-350786700: (630,-31590,-21600) [37,75,45,105]
-350786700: (-14220,-15930,14040) [38,12,126,51]
-350786700: (-9360,-33210,-1350) [38,30,90,105]
-350786700: (-18270,-36450,-3780) [67,84,126,78]
-350786700: (-16110,20070,10080) [77,51,81,-23]

-687541932: (-1638,-13734,286020) [1,3,549,1709]
-687541932: (7056,-16002,-64008) [7,84,0,254]
-687541932: (2520,-34146,-88956) [11,78,108,430]
-687541932: (2520,-41454,-34146) [22,54,6,211]
-687541932: (-9198,-49518,-10584) [29,42,126,210]
-687541932: (-15120,20034,27468) [33,18,156,-58]
-687541932: (4158,-37422,-39816) [33,99,57,145]
-687541932: (2520,-40194,-44352) [46,144,132,143]
-687541932: (-26208,-56322,-10584) [94,120,144,87]

-4297137075: (-48510,-65835,44100) [77,0,210,95]
-4297137075: (-66150,-34965,44100) [105,0,210,37]
-4297137075: (2520,-50715,-1023750) [1,54,132,6427]
-4297137075: (-3780,-123795,-160965) [3,27,501,5086]
-4297137075: (15750,-34965,-185220) [5,120,-90,537]
-4297137075: (30870,-34965,-94500) [7,168,-126,219]
-4297137075: (-17325,-107415,19530) [22,33,279,589]
-4297137075: (-28665,-145845,-73080) [74,159,243,277]
-4297137075: (630,-113085,-40950) [83,90,30,155]

====================================================

-5742252: (-702,-378,6084) [3,0,78,14]
-5742252: (360,-2574,-7362) [2,18,-6,137]
-5742252: (270,-3186,-6552) [3,18,6,122]
-5742252: (108,-3726,-7740) [3,18,24,154]
-5742252: (756,-1674,-4770) [3,27,-3,59]
-5742252: (-342,-4986,-5580) [7,27,51,101]
-5742252: (576,-3330,-2664) [10,24,0,37]
-5742252: (-558,-4986,-3420) [11,27,39,61]
-5742252: (-1908,2142,1656) [19,12,36,-10]

-86842700: (2100,-2310,-30380) [2,48,34,219]
-86842700: (1470,-11130,-23240) [7,42,14,186]
-86842700: (2800,-9590,-15050) [10,50,-10,101]
-86842700: (840,-14070,-18620) [11,42,28,154]
-86842700: (1750,-11410,-18620) [15,65,55,111]
-86842700: (210,-15750,-14840) [29,81,73,83]
-86842700: (-7280,9450,5880) [34,8,72,-29]
-86842700: (-5180,-18130,-3290) [37,37,59,61]
-86842700: (-6650,-20510,-6020) [47,67,79,61]

-1758564675: (630,-58275,-745920) [1,36,222,7363]
-1758564675: (6300,-22995,-188370) [5,90,120,751]
-1758564675: (4410,-42525,-196560) [7,84,126,843]
-1758564675: (4410,-54495,-130725) [13,87,81,526]
-1758564675: (-16380,-80325,-17955) [29,69,243,372]
-1758564675: (6300,-65205,-40635) [47,87,9,156]
-1758564675: (630,-72135,-28665) [53,63,21,154]
-1758564675: (-25830,-83475,-16380) [59,96,198,193]
-1758564675: (-40950,22995,28980) [89,48,162,-19]

-7034258700: (-116550,-23310,44100) [185,0,210,14]
-7034258700: (10080,-24570,-508410) [2,102,54,1671]
-7034258700: (7560,-54810,-598500) [3,90,60,2230]
-7034258700: (-23940,-6930,219870) [17,9,471,73]
-7034258700: (15750,-86310,-216720) [23,153,111,499]
-7034258700: (-39060,20790,132300) [37,30,360,-30]
-7034258700: (-40950,52290,112140) [41,3,333,-139]
-7034258700: (-29610,-132930,28980) [43,24,234,358]
-7034258700: (6300,-140490,-54180) [62,72,-6,251]

====================================================

-191575692: (-1260,-23814,1512) [2,12,234,1479]
-191575692: (5544,-12474,-18900) [11,66,-36,102]
-191575692: (-3528,-28854,-18270) [14,42,126,271]
-191575692: (-6552,-22554,2520) [22,24,108,127]
-191575692: (-2772,-29106,-24570) [22,66,108,183]
-191575692: (2520,-18774,-22050) [23,75,45,107]
-191575692: (3276,-20034,-13230) [25,51,-9,87]
-191575692: (-10710,-3654,13104) [31,27,123,25]
-191575692: (-1638,-26586,-20160) [41,99,93,97]

-4789392300: (-5040,-11970,705600) [2,0,840,665]
-4789392300: (-18900,-32130,176400) [15,0,420,238]
-4789392300: (-48510,-76230,44100) [77,0,210,110]
-4789392300: (630,-103950,-1413720) [1,39,297,12837]
-4789392300: (10080,-68670,-239400) [11,108,48,746]
-4789392300: (34650,6930,-103320) [11,198,138,206]
-4789392300: (-1260,-124110,-205380) [13,66,144,1142]
-4789392300: (9450,-69930,-250740) [15,135,195,713]
-4789392300: (30870,-76230,-69300) [37,141,-99,187]
-4789392300: (-35910,-51030,81900) [39,36,318,178]
-4789392300: (26460,-85050,-67410) [42,126,-84,197]
-4789392300: (-40950,18270,85680) [47,9,291,-37]
-4789392300: (-52920,-73710,42210) [69,99,303,167]
-4789392300: (-42840,-153090,-52920) [94,168,252,231]
-4789392300: (-10710,-125370,-31500) [101,81,57,143]
-4789392300: (-73080,-144270,-22050) [157,189,231,133]
-4789392300: (-88200,-82530,21420) [170,60,180,61]
-4789392300: (-98280,-1890,36540) [201,198,228,26]

===================================================

-39468627: (495,-7029,-34848) [2,27,39,449]
-39468627: (-594,-12771,-18810) [3,18,102,541]
-39468627: (-1584,-8811,6435) [5,9,111,218]
-39468627: (-396,-11781,-12870) [5,18,48,281]
-39468627: (2574,-4059,-9900) [5,48,-18,71]
-39468627: (-1881,-13563,-8712) [10,33,99,187]
-39468627: (198,-10395,-13068) [13,48,54,111]
-39468627: (-2178,-12573,-4554) [19,30,54,83]
-39468627: (1980,-8811,-5148) [19,36,-12,49]

-32618700: (360,-5850,-44190) [1,21,27,713]
-32618700: (-450,-10710,-9360) [5,15,45,253]
-32618700: (540,-7830,-16920) [6,36,42,173]
-32618700: (-2070,-9270,1440) [13,6,54,82]
-32618700: (2250,-3690,-9360) [13,66,54,62]
-32618700: (-3510,-8370,1980) [21,9,57,47]
-32618700: (-2700,-12690,-5850) [21,45,75,85]
-32618700: (-1800,-9990,-270) [34,6,18,33]
-32618700: (-7200,-10890,-720) [43,72,96,46]

-48726700: (-1430,-1650,25080) [3,1,159,67]
-48726700: (-110,-12650,-31460) [3,23,71,529]
-48726700: (220,-10890,-31350) [3,25,45,445]
-48726700: (2860,-4070,-11330) [9,59,23,67]
-48726700: (-1100,-13310,-7040) [10,20,50,159]
-48726700: (1210,-9130,-12980) [11,44,22,102]
-48726700: (1320,-8910,-12650) [14,54,38,87]
-48726700: (-7590,-6270,3520) [41,6,62,18]
-48726700: (-8360,-2530,4180) [43,38,76,14]

-3946862700: (-21780,-48510,108900) [22,0,330,245]
-3946862700: (4950,-18810,-580140) [1,72,78,2714]
-3946862700: (-3960,-94050,189090) [2,18,714,5939]
-3946862700: (-5940,-50490,391050) [3,9,669,2093]
-3946862700: (19800,-30690,-137610) [5,135,-105,367]
-3946862700: (-15840,-133650,-95040) [29,96,288,618]
-3946862700: (-24750,-107910,1980) [35,60,270,394]
-3946862700: (-32670,-92070,25740) [51,36,222,218]
-3946862700: (-87120,-38610,29700) [163,30,180,30]

Total computational time: about 10 minutes
*