Joint research 2002

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, Québec, 1988
D. C. Mayer, Multiplicities of dihedral discriminants, Math. Comp. 58 (1992), 831-847 and S55-S58
E-mail addresses:

Pure cubic fields ordered by ascending conductors (2002/03/03)
Dan (02/03/03): Today I start a series of reports
featuring one of my favourite topics: the pure cubic fields.

Ordering pure cubic fields L = Q((a*b^2)^1/3)
with respect to ascending conductors f = [3*]a*b
yields other statistics of Dedekind Types (DT) and Principal Factorization Types (PFT)
than ordering them by ascending radicands R = a*b^2.

According to the Theorem of Hasse,
the relation between the conductor f and the discriminant d of a pure cubic field L
is given by d = f^2*d_0 = -3f^2.
Ordering pure cubics by increasing conductors is therefore the same as
ordering them by increasing absolute values of discriminants.
Dan (02/03/03):
I have just computed the 7811 pure cubics with conductors f < 10000.

Using appropriate database queries I found the following statistics:

Population of the Dedekind Types:
DT 1A (v_3(f) = 2): 1999 fields (26 %)
DT 1B (v_3(f) = 1): 2198 fields (28 %)
DT 2 (v_3(f) = 0): 3614 fields (46 %)

Population of the Principal Factorization Types:
PFT ALPHA (relatively ambiguous principal ideals): 1830 fields (23 %)
PFT BETA (absolutely ambiguous principal ideals): 5609 fields (72 %)
PFT GAMMA (some unit in the normal field has norm [-1+(-3)^1/2]/2): 372 fields (5 %)

Due to the accumulation (46% instead of 23%) of DT 2 in lower conductor ranges
(since DT 2 has f = a*b, whereas DT 1 has f = 3*a*b),
we observe a slight overall increase of PFT Alpha (23% instead of 20%)
and PFT Gamma (5% instead of 3%),
since DT 2 has a considerably lower percentage (55%) of PFT Beta than DT 1 (83%).
Dan (02/03/03):
However, it should be pointed out that
there are only 4708 conductors in the range f < 10000,
since they arise in complete Families of fields (L_i)_{1 <= i <= m}
with various Multiplicities m.

Using SQL database queries of the type
"SELECT * INTO CondHexadekPlus FROM CondExt
WHERE ( CondExt.f In
(SELECT f FROM CondExt AS Temp GROUP BY f HAVING Count(*) > 16) )
ORDER BY CondExt.f, CondExt.D;"
I found the following distribution of Multiplets:

2945 Singulets (m = 1)
1164 Doublets (m = 2)
231 Triplets (m = 3)
264 Quadruplets (m = 4)
13 Quintuplets (m = 5)
10 Sextuplets (m = 6)
79 Octuplets (m = 8)
2 Hexadecuplets (m = 16)

Multiplicities m = 10,11,12 will occur for bigger conductors f,
but m = 7,9,13,14,15 are impossible for theoretical reasons.

See D. C. Mayer, Multiplicities of dihedral discriminants,
Math. Comp. 58 (1992), 831-847 and S55-S58
Dan (02/03/03):
This was only a piece of cake.
It's a small byproduct of the most extensive computation of pure cubics
of all times.

Tomorrow, I'll bring the Lion !!! Wow !!!
Dan (02/03/04): Historical Remark:
The above mentioned database extends the results of

[1] Ken Nakamula, A Table for Pure Cubic Fields,
Advanced Studies in Pure Math. 13 (1988), 461-477,

which was the only table of pure cubic fields
ordered by ascending conductors, up to now.
It lists for each of the 358 fields with conductors 10 <= f <= 618
(corresponding to discriminants -300 >= d >= -1145772
and excluding the conductors f = 6,9),
the conductor f,
the linear and quadratic part a,b of the radicand R = a*b^2,
the class number h and an upper bound B for h,
and the coefficients of the minimal polynomial
M_e(X) = X^3 - Tr(e)X^2 + Tr(1/e)X - 1
of the fundamental unit e.

This table had already previously appeared in

[2] Ken Nakamula, Elliptic units and the class numbers,
Dissertation, Tokyo Metropolitan University, 1985.

The computational method was developed theoretically in

[3] Ken Nakamula, Class number calculation of a cubic field from the elliptic unit,
J. Reine Angew. Math. 331 (1982), 114-123.

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