Joint research 2002

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


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On these pages, we present most recent results of our joint research, directly from the lab.
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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, Québec, 1988
D. C. Mayer, Multiplicities of dihedral discriminants, Math. Comp. 58 (1992), 831-847 and S55-S58
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Web master's e-mail address:
contact@algebra.at
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Families of pure cubic fields (2002/03/18)
Dan (02/03/18):
Now I have ordered the 827600 pure cubic fields L = Q(R^1/3)
with normalized radicands R = a*b^2 < 10^6
by increasing conductors f = [3*]a*b < 3*10^6.

This database is more than 100 times bigger than the previously mentioned
excerpt containing the 7811 pure cubics with conductors f < 10^4.
Filtering out this excerpt was by no means random but had a very precise reason:

I have a general theorem warranting that, for any upper bound B,
an excerpt of the pure cubic fields with conductors f < B^{2/3}
from a table of all pure cubic fields with radicands R < B
consists of Complete Families.
And taking B = 10^6 obviously yields B^{2/3} = 10^4.

Without filtering this excerpt, however,
an overwhelming part of the families is incomplete.
It is illuminating to separate the occuring
formal families of fields (L_i)_{1 <= i <= n}
with various formal multiplicities n <= 16
into complete and incomplete families,
corresponding to diagonal (boldface) and subdiagonal entries
in the following lower triangular matrix:
n=1n=2n=3n=4n=5n=6n=7n=8n=9n=10n=11n=12n=13n=14n=15n=16
m=1219834000000000000000
m=21724392690200000000000000
m=3511561143527430000000000000
m=4615361865453984672000000000000
m=511075216350123711600000000000
m=6146833953640274105920000000000
m=828277813554362732109445242651500000000
m=10150122445181270200000000
m=11110730235271243210100000
m=127001901510220010000000
m=167817209716144564421931208252343114217828
m=2013000000000000000
m=2118000000000000000
m=2243800000000000000
m=241000000000000000
m=327441851192122157412112001
m=648000000000000000

Obviously there are cumulations of families
at 2-power multiplicities m = 4,8,16,32,
which remind me of nuclear islands of stability (magic numbers of nucleons).
Dan (02/03/18):
Nevertheless, we now have much more Complete Multiplets, too:

219834 instead of 2945 Complete Singulets (m = 1)
26902 instead of 1164 Complete Doublets (m = 2)
2743 instead of 231 Complete Triplets (m = 3)
4672 instead of 264 Complete Quadruplets (m = 4)
116 instead of 13 Complete Quintuplets (m = 5)
92 instead of 10 Complete Sextuplets (m = 6)
515 instead of 79 Complete Octuplets (m = 8)
1 instead of 0 Complete Hendecuplets (m = 11)
28 instead of 2 Complete Hexadecuplets (m = 16)

True multiplicities m = 10,12,20,21,22,24,32,64 occur only for incomplete families,
m = 40,42,43,44,... will occur in higher ranges of conductors f, and
m = 7,9,13,14,15,17,18,19,23,25,26,27,28,29,30,31,33,34,35,36,37,38,39,41,...
are impossible for theoretical reasons.

See D. C. Mayer, Multiplicities of dihedral discriminants,
Math. Comp. 58 (1992), 831-847 and S55-S58

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