MATHEMATICS OF COMPUTATIONVOLUME 50, NUMBER 182APRIL 1988, PAGES 581-594
On Totally Real Cubic Fields
with Discriminant D < 107
By Pascual Llórente and Jordi Quer
Abstract. The authors have constructed a table of the 592923 nonconjugate totally
real cubic number fields of discriminant D < 107, thereby extending the existing table
of fields with D < 5 x 105 constructed by Ennola and Turunen [4]. Each field is given by
its discriminant and the coefficients of a generating polynomial. The method used is an
improved version of the method developed in [8]. The article contains an exposition of
the modified method, statistics and examples. The decomposition of the rational primes
is studied and the relative frequency of each type of decomposition is compared with
the corresponding density given by Davenport and Heilbronn [2].
1. Introduction. A table of totally real cubic fields with discriminant D < D'
has been constructed by Godwin and Samet [5] for D' = 2 x 104. Angelí [1]
extended this table up to D' = 105 by using a similar method. These tables are not
complete (see [4] and [9]). A complete table for D' = 105 is constructed in [8] by a
different method. Finally, a third method is developed by Ennola and Turunen [4]
to compute a table with D' = 5 x 105. In this paper we shall describe an extended
table for D' = 107. The method used is an improved version of that developed in
[8]. Up to 5 x 105, this table agrees with Ennola and Turunen's.
The modified method and the new algorithm are described in Sections 2 and
3. Section 4 contains statistics and examples (Tables 1-6). The decomposition of
the rational primes is studied using the congruential criteria given in [6] and [7].
In Section 5 we compare the relative frequency of each type of decomposition for
different primes with the corresponding density given by Davenport and Heilbronn
in [2] (Tables 7-11).
Computations were done on an IBM 3360 owned by the Universität de Barcelona
and a VAX-8600 owned by the Facultat d'Informàtica de Barcelona.
2. The Improved Method. The method used to compute our table of totally
real noncyclic cubic fields is similar to the one described in [8] but improved in
several ways. In this section we recall the main ideas in [8] and we explain the
modifications introduced. For every prime p G Z and integer m, vp(m) denotes the
greatest integer k such that pk divides m.
Each triple of conjugate noncyclic fields is defined by a polynomial of the type
(1) f(a,b,X) = X3 -aX + b,
Received September 22, 1986; revised February 17, 1987 and August 7, 1987.
1980 Mathematics Subject Classification (1985 Revision). Primary 11R16, 11-04.
Key words and phrases. Real cubic fields.
©1988 American Mathematical Society
0025-5718/88 $1 00 + $.25 per page
581
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
582 PASCUAL LLÓRENTE AND JORDI QUER
where a and b are positive integers such that
(2) f(a, b, X) is irreducible in Q[X],
(3) there is not a prime p with vv(a) > 2 and vp(b) > 3.
Then, each field on the triple is isomorphic to the cubic field K = K(a,b) = Q(6),
where 9 is a root of the polynomial f(a, b, X). The descriminant of f(a, b, X) is
(4) D(a, b) = 4o3 - 2762 = DS2,
where D — D(K) is the descriminant of the cubic field K and S > 0 is the index
of 9. It is known (cf. [3] or [7]) that
(5) D = D(K)=dT2,
where d is the discriminant of Q(VD) and T = 3mT0 with 0 < rn < 2 and T0 > 0
is a square-free integer having no common factor with 3d. Note that d > 1, since
we are assuming that K is totally real and noncyclic.
Consider the congruences
(6) a = 3 (mod 9), b = ±(a -1) (mod 27).
We have the following result (cf. [3, p.112]).
THEOREM 1 (VORONOI). (i) If the congruences (6) are not satisfied, then S
is the greatest positive integer whose square divides D(a, b) for which there exist
integers t, u and v such that
(7) -S/2<t<S/2, Zt2~a = uS, t3 -at + b = vS2,
and 1,9, ß, with ß = (92 +t9 + t2 — a)/S, is a basis for the integers of K.
(ii) // the congruences (6) are satisfied, then S = 27S', and S' is the greatest
positive integer whose square divides D(a,b)/729 for which there exist integers t,u
and v such that
(8) -3S'/2<t<3S'/2, 3t2-a = 9uS', t3 - at + b = 27vS'2,
and 1, ip, ß, with tp — (9 - £)/3 and ß = (92 +t9 + t2 - a)/9S, is a basis for the
integers of K.
It follows that by choosing a minimal t in (7) or (8) there is a unique quadruple
(S,t,u,v) of integers associated with each pair (a, b). The main idea in [8] is to
associate a positive definite binary quadratic form F(a,b) with each pair (a, b),
whose coefficients are given in terms of a, S, t, u and v. We shall give an alternative
definition for the quadratic form associated with the pair (a, b) (see D, below). We
repeat here the definition and some of the properties of F(a, b) given in [8], to make
our exposition more self-contained.
If the congruences (6) are not satisfied, then Tr(l) — 3, Tr(0) = 0 and Tr(/3) = u
are the traces of the integers in a basis for the integers of K. So the integers 7 € K
with zero trace are given by
(9) 7 = (Zx92 + Z(xt + Sy)9 - 2ax)/SS; x,yeZ, 3 | ax.
Let 7 t¿ 0 be such an integer. Its minimum polynomial is f(a', —N(~i),X), N(-y)
being the norm of 7 and a' = (3u2 - 27tv)x2/9 + (Zut - 9vS)xy/3 + ay2.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
ON TOTALLY REAL CUBIC FIELDS WITH DISCRIMINANT D < 107 583
If 3 f u, since Tr(9) = 0, we have 3 | x. Let z = x/3; then
a' = (3u2 - 27tv)z2 + (3ut - 9vS)zy + ay2.
If 3 | u, let u = 3w; then
a' = (3w2 - 27tv)x2 + (ut - 3vS)xy + ay2.
If the congruences (6) are satisfied, then
a = 3oi with ai = 3a2 + 1 and a2 €E Z,
ct = 3ui+n with |r/| < 2 and ui,n G Z,
t = 3ii + 6 with 6 = ±1 and t\ € Z.
The integers 7 € Ä" with zero trace are given by
(10) 7 = (x92 + (tx + 3fiSx + 9Sy)9 - 2axx)/9S; x,y&Z, p = ne.
Let 7 t¿ 0 be such an integer. Its minimum polynomial is f(a', -N(^),X), N(i)
being the norm of 7 and
a' = (uui + 2nui —vt + puti - 3pvS' + u2a2 + p2)x2 + (ut + 2pai - 9vS')xy + ay2.
With these notations, the quadratic form F(a, b) associated with the pair (a, b)
is defined in [8] as follows:
DEFINITION 1. (i) If the congruences (6) are not satisfied and 3 \ u, we define
F(a, b) = (3u2 - 27tv, 3ut - 9vS, a).
(ii) If the congruences (6) are not satisfied and u = 3w, we define
F(a,b) = (3w2 - 3tv,ut - 3vS,a).
(iii) If the congruences (6) are satisfied we define
F(a, b) = (mti + 2nui — vt + puti — 3uvS + u2a2 + p2,ut + 2pai — 9vS, a).
Then F(a, b) represents precisely those integers a' > 0 for which there exists an
integer b' such that K(a',b') is isomorphic to K(a,b). Then, if K(a,b) «a K(a',b'),
the corresponding associated forms F(a, b) and F(a', b') represent the same integers
and, consequently, they are equivalent (cf. [8] or [10]).
THEOREM 2. Let F — F(a, b) be the form associated with the field K = K(a, b)
of discriminant D — D(K). Then the discriminant D(F) of the form F is
D(F) = -D/3 if 27 \D,
D(F) = -3D if 27 f D.
Proof. One computes D(F) from Definition 1. Then D(F) = -D/3 in case (ii),
and D(F) = —3D in cases (i) and (iii).
In case (ii) of Definition 1, we have a = 3ai (since u = 3w) and then 27 | DS2
in (4). If 3 f S then 27 | D; else, easy congruential considerations show that if
3 I u and 9 | S, then the congruences (6) are satisfied, hence we must have S = 3Si
with 3 f Si and D = 3Di. From (7) it follows that 01 = t2 ( mod 3) and b = 2t3
(mod 3). Then DiSf = 4a? - b2 = 0 ( mod 3), so 9 | D and from Theorem 2 of [7],
27 must divide D.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
584 PASCUAL LLÓRENTE AND JORDI QUER
Conversely, from Theorem 2 of [7] and easy congruential considerations one can
show that if 27 | D then case (ii) of Definition 1 holds. D
From this result and the elementary theory of reduced positive definite quadratic
forms we obtain
THEOREM 3. Let K be a totally real noncyclic cubic field of discriminant D =
D(K). Then K «s K(a,b) for some integer a with
a < y/D/3 if 27 | D,
a<\ÍD if 27 \ D.
In this case, we have
S <S(a) = 2s/a~¡3 if 27 \D,
S<S(o) = 2v/ö" if 27 i D.
As a consequence, a table of all totally real noncyclic cubic fields of discriminant
D = D(K) < D' can be constructed from a finite number of pairs (a,b), carrying
out the following steps:
- Elimination of all pairs not satisfying (2) or (3).
- Decomposition of D(a, b) as in (4) for the remaining pairs.
- Elimination of all pairs not satisfying the bounds of Theorem 3.
- Elimination of isomorphic fields.
In this way, a table with D' = 105 was constructed (see [8] and [9]); but this
algorithm is too inefficient (about 70 hours of computer time were needed to con-
struct that table) to be applied to much greater D'. Moreover, the computation of
D(a, b) requires multiple-precision arithmetic. We explain below the improvements
introduced in the method. With them, the method becomes much more efficient.
A. Irreducibility of f(a, b, X). It is convenient to observe that each of the follow-
ing conditions implies the irreducibility of f(a,b,X) in Q[X]:
(11) 1 < vp(b) < Vp(a) for some prime p,
(12) a = b = 1 ( mod 2),
(13) a = 1 ( mod 3) and 3 + 6.
In [8] one uses the fact that f(a,b,X) is reducible if and only if it has a root
m G Z. In this case, m | b, and an elementary study of f(a,b,X) shows that such
a root m must satisfy
0 < m < y/a~ or \/E < —m < \/ä + \Ja/3.
B. Computation of D(K) and S. In [8], D = D(K) and S were computed from
D(a, b) using Theorem 1. This was certainly the most laborious part of the method.
The results in [7] simplify this computation. Indeed, if (11) is satisfied for a prime
p, then p | T and (11) is also a necessary condition for p | T if p ^ 3; the factor
3m of T can be determined from certain congruential conditions involving a and b
(see [7, Theorems 1 and 2]). In this way, T is computed and, eventually, a divisor
So of S is also obtained. It is also easy to compute v2(D) and v2(S) from a and
b in case 2 \T. Now, every new integer m whose square divides D(a,b) will be
a new factor of S, i.e., m | S if and only if D(a,b) = 0 ( mod m2). Also, these
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
ON TOTALLY REAL CUBIC FIELDS WITH DISCRIMINANT D < 107 585
computations sometimes give several prime factors of d; in fact, p \ d if any of the
following conditions is satisfied:
(14) 1 = vp(a) < vp(b),
(15) vp(D(a,b)) is odd.
These conditions are also necessary for p \ d, except if p = 3. In this case, some
additional congruential conditions must be considered (see [7, Theorem 1]). Using
the bound S < S(a) in Theorem 3, one can easily compute S (and then D) or
eliminate the pair (a,b).
C. Eliminating Superfluous Fields. With the help of Theorem 3 we can eliminate
all pairs (a, b) for which S > S(a). We can eliminate also the pairs with D < D(a),
where
(16) D(a)=9a2 or D(a) = a2, according as 27 | D or not,
because in this case K(a,b) ss K(a',b') for some a' < a. The determination of the
bounds S(a) and D(a) (i.e., if 27 | D or not) follows from the computations in B.
In many cases, this elimination can be done by knowing only some factors of D and
S, and it is not necessary to compute S and D completely as in [8].
D. Eliminating Isomorphic Fields. If we obtained two fields Kx = K(ai,bi) and
K2 = K(a2,b2) with the same discriminant D, we have to test if Kx sa K2 or
not. As in [8], this can be done by studying their associated quadratic forms. The
method has been improved at this point too. We start giving a new definition of
the associated quadratic form:
DEFINITION 2. (i) In the first case of Voronoi's Theorem, let B - (3b-2at)/S.
Then we define
F* = F*(a,b) = (a,B,C) if 27 | D,
F* = F*(a,b) = (a,3B,C) if 27 \ D,
where C is determined by the condition D(F*) = -D/3 or D(F*) — -3D, accord-
ing as 27 | D or not.
(ii) In the second case of Voronoi's Theorem, let a' — a/3, let w be the only integer
with |u| < 2 and « = t(t2 - a')/3S' ( mod 3) and let B = -2pa' + (b - 2a't)/3S'.
Then we define
F* = F*(a,b) = (a,B,C),
where C is determined by the condition D(F*) = -3D.
It is immediate to see that the associated form F*(a,b) given in this definition
is equivalent to the associated form F(a,b) given in Definition 1. An advantage
of this new definition is that it is not necessary to compute u and v in Voronoi's
Theorem. Moreover, if a is minimum, the reduced equivalent quadratic form is
easily obtained from a and B; it suffices to find n G Z such that
(17) B' = B + 2an and \B'\<a.
Then the reduced form will be F' — (a,B',C), where C" is determined by the
discriminant.
The following theorem permits us to eliminate a large number of isomorphic
fields.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
586 PASCUAL LLÓRENTE AND JORDI QUER
THEOREM 4. Let Ki = K(ai,bi) satisfy the conditions of Theorem 3, and let
F' = (ai,B',C) be the reduced form equivalent to its associated form. Then there
exists a pair (a2,b2) with a2 > ai, and b2 ^ bi in case a2 = ai, such that the field
K2 = K(a2,b2) satisfies the conditions of Theorem 3 and is isomorphic to Ki if
and only if D(C) < D(Ki). In this case, we must have C = a2, and there is only
one such pair.
Proof. We know that a2 = F'(x,y) must be an integer represented by F', and
by (3) we have that y ^ 0. It is easy to see that C" = .F(0,1) is the only integer
a represented in that way by F' for which the condition D(a) < D(Ki) can be
satisfied. D
Theorem 4 is not sufficient to eliminate isomorphic fields if there exists a third
field A3 = K(ü3,bs) satisfying the conditions of Theorem 3, with the same dis-
criminant as Ki and K2 and with a2 = a^ = C (D = 77844 is an instance of this
case). Then we have to decide whether Ki « K2 or Ki sa ä"3. For this we can
proceed as in [8]: The representation of a2 by F(ai,bi) determines an integer 7 in
Ki with zero trace, whose minimal polynomial is f(a2, —N(^),X). Then Ki k, K2
or Ki sa A3 according as |iV('y)] is equal to b2 or 63.
Using the definition of 7 (see (9) and (10)) and the transformation taking
F(ai,6i) into its reduced associated form F' = (a1,5',a2), and computing ex-
plicitly the norm N(*i) of the integer 7, we obtain
THEOREM 5. Let Ki = K(ai,bi) satisfy the conditions in Theorem 3 with
D(Ki) = D. Let F' = (ai,B',a2) be the reduced form equivalent to its associated
form F — F(ai,bi) with D(a2) < D, and let 7 be the null trace integer in Kx
determined by the representation of a2 by F. Then:
(i) IfD = 27D' and ax = 3a, then
N{i) = ((2aa2 - D')S2 - 4a3 +bx(t- nS)(ai - (t - nS)2))/S3.
(ii) If 27 ^ D and the congruences (6) are not satisfied, then
N{i) = ((2axa2 - D)S2 - 4a? + 61 (3* - r»S)(9ai - (3i - nS)2))/S3.
(iii) If ai = 3a and the congruences (6) are satisfied, then
N(1) = ((2a1a2-D)27S"2-4a3+bi(t+3uS'-9nS')(ai-(t+3uS'-9nS')2))/(9S')3.
Here, S, S', t and u are the integers used in Definition 2 and n is the integer deter-
mined by (17).
3. The New Algorithm. The method described in the previous section pro-
vides an easy computer-programmable algorithm to construct a table of the totally
real cubic fields K with discriminant D = D(K) < D' for a given D'. We shall
now describe the algorithm used by the authors.
We take all integers a with 4 < a < y/TP and, for each, we take the integers b
with 1 < b < 2(a/3)3//2. For each pair (a, b) we proceed as follows:
Step 1. Compute M = g.c.d.(a,6) and work with every prime factor p of M.
During these computations:
(a) The pairs not satisfying (3) are eliminated.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
ON TOTALLY REAL CUBIC FIELDS WITH DISCRIMINANT D < 107 587
(b) T0 is always obtained and T is also obtained if 3 | M (as is explained in B of
Section 2). In case 3 | M, the bounds S (a) and D(a) are determined.
(c) vp(D) and vp(S) are computed using (b) and (14).
(d) In some cases, irreducibility of f(a,b,X) is proved by (11).
Step 2. If irreducibility of f(a,b,X) has not been proven in Step 1(d), use the
other results in A of Section 2 to eliminate the pairs with /(a, b, X) reducible.
Step 3. If 3 -f M, compute v$(D) and t>3(S), using the congruential conditions
given in [7]. So, T is obtained and the bounds S(a) and D(a) are determined.
(Remark. During the computations, divisors So of S and Do of D are obtained.
If So > S(a) or Do > D', the pair (a,b) is eliminated.)
Step 4. If 2 f M, compute v2(D) and v2(S), using the congruential criterion
given in [7].
Step 5. Let Si = S (a)/So- For every prime p with p \ 6M and p < Si, examine
D(a,b) modulo p2 (if p2 < Si then examine D(a,b) modulo p4). In this way:
(a) Vp(D) and vp(S) are computed using (15).
(b) Eventually, Do, So and Si are modified.
(c) The final S0 is the greatest S in (4) with S < S (a).
Step 6. Let Di = D(a,b)/S(2. If Dx > D', the pair (a, b) is eliminated. Indeed,
in this case we have D > D' or S > S (a).
Step 7. Let D2 = Di/D0T2. The pair (a, b) is eliminated in the following cases:
(a) If D2 is not square-free (in this case, S > S(a)).
(b) If D2 = D0 = 1 (in this case, d = 1).
(c) IfDi <D(a).
Step 8. If the pair (a, b) has not been eliminated in the preceding steps, K =
K(a,b) is a cubic field with discriminant D = Di < D' satisfying the conditions of
Theorem 3. During the process, d — D0D2, T and S = Sr, have been computed.
Record these data in a file.
Step 9. Data in that file are ordered with increasing discriminant D without
altering the order of their generation among the fields with the same discriminant.
Step 10. Eliminate isomorphic fields in the file by using the results of D in
Section 2. To do this, proceed as follows: Let Ki = K(ai,bt), i = 1,...,N,
be all the fields in the file with the same discriminant D. According to Step 9,
we have ai < a2 < • ■ • < a^. Compute the reduced quadratic form (ai,i?,C)
equivalent to the quadratic form associated with the pair (ai,6i). By Theorem 4,
Ki is isomorphic to some Ki with i > 1 (and only to one of them) if and only if
D(C) < D, and in this case, C = a,. If C = ai for only one t > 1, then eliminate
the field Ki. If C = a¿ for more than one i > 1, then compute N(i) by using
Theorem 5 and eliminate the field Ki = A"(a¿,6¿) for which í>¿ = |JV(7)|. Proceed
in the same way with the next noneliminated field until the set of all fields with
discriminant D is completely purged of isomorphic pairs.
This algorithm is very efficient. A table for D' = 105 can be constructed in less
than a minute of computer time, and with D' = 5 x 105 in about five minutes.
We have used it with D' = 107, and the total computer time required was about 5
hours. Almost all time was spent in the generation of the fields K(a, b) (Steps 1 to
8). The purge of isomorphic fields in Step 10 eliminated around 10% of the stored
fields and required about 8 minutes.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
588 PASCUAL LLÓRENTE AND JORDI QUER
4. Totally Real Cubic Fields with Discriminant D < 107. The table con-
taining the 592422 nonconjugate totally real noncyclic cubic fields with discriminant
less than 107 consists of 10 sectors, the kth containing the fields with discriminants
between I06(k — I) and 106fc. For each field K = Q(9), its discriminant D, the
coefficients a and 6 of a polynomial Irr(#, Q) = f(a, b, X) and the integers 5 and T
are listed. Other data obtained during the computations were not stored for lack
of space. Separately, we have constructed a table of the 501 cyclic cubic fields with
discriminant less than 107.
TABLE 1
Number of cubic fields with discriminant 0 < D < 107
Sector
1'2
34
5IS
7
8
910
Noncyclic
54441
577775878759266597385999460376605076070560831
Cyclic
159
6747
44
36
3627
352525
Total
54600
57844588345931059774
6003060403
605426073060856
Noncyclic
(accum.)
54441112218171005230271290009350003410379470886531591592422
Cyclic
(accum.)
159226273317353389416451476501
Total
(accum.)
54600112444171278230588290362350392410795471337532067592923
TABLE 2
Number of discriminants 0 < D < 107 with 2 associated nonconjugate cubic fields
Sector
1
234
56789
10
Noncyclic
166
223206218231221
241224
262239
Cyclic
37
18
10
11
999
69
Total
203
241
216
229240230250230271247
Noncyclic
(accum.)
166
389595813
104412651506173019922231
Cyclic
(accum.!
37
556576
8594
103109
118
126
Total
(accum.;
203
444
660889
1129
1359160918392110
2357
Table 1 gives the number of cubic fields K by sectors. We give for each sec-
tor k the number of noncyclic cubic fields, cyclic cubic fields and cubic fields with
discriminant 106(fc - 1) < D < 106/c and with discriminant 1 < D < 106fc (accu-
mulated). Tables 2, 3 and 4 are similar for the number of discriminants with N
associated nonconjugate fields, for N — 2, 3 and 4, respectively.
There are five discriminants D < 107 with N associated nonconjugate fields
for N > 4. And we have N = 6 for all of them. In Table 5 we have listed
these discriminants, their decomposition D — d32mTr2 and the coefficients a and
b of the polynomials f(a,b,X) defining the six corresponding fields. Among the
discriminants corresponding to noncyclic fields in Table 4, there are nine with T > 1;
these discriminants are listed in Table 6 in a way similar to those in Table 5.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
ON TOTALLY REAL CUBIC FIELDS WITH DISCRIMINANT D < 107 589
TABLE 3
Number of discriminants 0 < D < 107 with 3 associated nonconjugate cubic fields
Sector
1
2
34
56
789
10
Noncyclic
343468557552
568601
624622582626
Noncyclic
(accum.)
343811
13681920248830893713433549175543
TABLE 4
Number of discriminants 0 < D < 107 with 4 associated nonconjugate cubic fields
Sector
1
2
34
56789
10
Noncyclic
161
233
255
284283
293
311
348364347
Cyclic Total
162234
256285284
294
311350364347
Noncyclic
(accum.)
161394649933
121615091820
216825322879
Cyclic(accum.
Total(accum.)
162
396652937
1221
151518262176
25402887
From Tables 4 and 6 we observe that there are 2879 discriminants D = d,
1 < D < 107, with four associated noncyclic fields. From class field theory we can
conclude that there are 2879 real quadratic fields with discriminant d < 107 whose
ideal class group H(d) has 3-rank equal to 2. A complete study of the H(d) having
3-rank > 1 for d < 107 will be given in a later paper.
5. On Davenport and Heilbronn's Densities. The total number of noncon-
jugate cubic fields with discriminant less than 107 is 592923, giving the empirical
density 0.05929. Table 1 easily permits the computation of this empirical density
in each sector. Davenport and Heilbronn prove in [2] that the asymptotic value
is (12c(3))_1 « 0.06933. Thus the convergence is very slow, as was noted in [11].
In [2], Davenport and Heilbronn obtain also the asymptotic value of the density
of each type of decomposition of a rational prime p in cubic fields. The values
obtained for them are:
l/w for p = P3,
p/w for p = PQ2,
p2/3w for p = P,
p2/2w for p = PQ,
p2/6w for p = PQR,
with w = p2 +p+ I.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
590 PASCUAL LLÓRENTE AND JORDI QUER
Table 5
Discriminants 0 < D < 107 with 6 associated cubic fields
D=32"T d
Coefficients of the polynomial f(a,b,X~)
3054132 84837
96102150
210336366
134210
622112212442674
4735476 9 2 131541
108114168252288648
106210726
129218344772
5807700 81 10 717
180270360450540720
60990
246028504740
6690
6367572 81 2 19653
126
342396450540720
246174
2994364238523012
9796788 81 2 30237
144216540648756918
282204
320452207794
5742
Using the results of [6] and [7], we have computed the decomposition type of the
rational primes p for 2 < p < 181 in each of the 592422 noncyclic cubic fields in
the table. In particular, there are 46532 noncyclic cubic fields K with D(K) < 107
having 2 as its common index divisor, i.e., with the rational prime 2 decomposing
completely. We give in Tables 7, 8, 9 and 10 the empirical density of each type of
decomposition by sector and its asymptotic value for p = 2,3,5 and 7, respectively.
In Table 11 we give the empirical density and its asymptotic value of each type of
decomposition for the primes p with 2 < p < 100 in all fields in the table.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
ON TOTALLY REAL CUBIC FIELDS WITH DISCRIMINANT D < 107 591
TABLE 6
Discriminants 0 < D < 107 with 4 associated cubic fields and T = 32mTo > 1
D=3*"Td 32"' T,
Coefficients of the polynomial f(a,b,X)
1725300 51 10 21390
180270360
210780
15302580
2238516 81 14 141126252252378
462546
14282814
2891700 il 10 35790
180180360
30660870
1290
4641300 81 10 573180270540540
420117015904140
6810804 51 14 429126378378504
210130223944326
7557300 81 10 933270450720810
6302550
6608730
7953876 81 14 501126252378756
421092
7987308
8250228 9 14 4677336630756840
1694266
79248764
8723700 51 10 1077270450540900
90363034208700
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
592 PASCUAL LLÓRENTE AND JORDI QUER
TABLE 7
Type of decomposition of the prime 2 in the
noncyclic cubic fields with discriminant 0 < D < 107 (%)
Sector PJ PQ2 PQ PQR
1
234
56
789
10
15.79715.37815.17315.16715.21115.04815.11515.03814.90015.163
27.54027.80027.99627.82428.00428.01428.0282802328.15327.951
21.45820.86020.61220.59020.45920.46720.41220.37920.33120.210
28.33528.38528.51028.50428.41228.44328.42928.47128.44628.532
6.8707.5777.7097.9157.9138.0278.0168.0888.1718.144
All Table 15.192 27.938 20.567 28.448 7.855
Theoretical 14.286 28.571 19.048 28.571 9.524
TABLE 8
Type of decomposition of the prime 3 in the
noncyclic cubic fields with discriminant 0 < D < 107(%j
Sector PQ2 PQ PQR
1
2
34
567
89
10
8.4168.154
8.1628.0608.1518.0828.1098.0557.881
8.162
22.25722.447
22.49322.54622.66122.61222.65822.53822.652
22.628
25.98825.27125.03824.94324.84224.74624.71024.66324.65924.458
34.71834.69434,68334.68334.58434.71034.46934.76834.816
34.670
8.6209.4359.6259.7689.7639 849
10.054
9.9769.993
10.082
All Table 8.120 22.553 24.919 34.679 9.729
Theoretical 7.692 23.077 23.077 34,615 11.538
TABLE 9
Type of decomposition of the prime 5 in the
noncyclic cubic fields with discriminant 0 < D < 107 (%)
Sector PQ2 HQ PQR
1
2
34
56
789
10
3.4773.4153.3753.3903.2963.4343.3793.34233113.362
15.37315.65015.70215.66515.79715.75215.79315.78515.67315.869
30.14829 23029.10328.82128.85128.67628.72828.64128.58728.558
40.55840.53340.47840.48740.55240.38740.40240.38440.552
40.307
10.444
11.17211.34111.63711.50411.75111.69811.84811.877
11.905
All Table 3.377 15.709 28.920 40.463 11.531
Theoretical 3.226 16.129 26.882 40.323 13.441
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
ON TOTALLY REAL CUBIC FIELDS WITH DISCRIMINANT D < 107 593
TABLE 10
Type of decomposition of the prime 7 in the
noncyclic cubic fields with discriminant 0 < D < 107 (%)
Sector PQ2 PQ PQR
1
2
3
45
6
78
910
1.8591.784
1.8291.7941.8631.7841.8471.7781.8171.863
11.71211.87311.95511.86311.86812.01011.95312.05111.93111.843
31.99230.96730.86930.67530.58730.65630.55530.13430.50330.513
43.14843.34142.99243.36643.05342.98443.03543.39343.07142.942
11.28912.03412.35512.30212.62812.56612.60912.64312.67812.840
All Table 1.822 11.908 30.731 43.131 12.408
Theoretical 1.754 12.281 28.655 42.982 14.327
TABLE 11
Type of decomposition of the rational primes p < 100 in the
noncyclic cubic fields with discriminant 0 < D < 107 (%)
Prime
EmpiricalTheore-
tical
PQ2
EmpiricalTheore-
ticalEmpirical
Theore-
tical
PQ
EmpiricalTheore-
tical
PQR
EmpiricalTheore-
tical
2
3
57
1113
17
19
23
2931
37
41
43
47
535961
67
71
7.3
79
83
8997
15.192
8.1203.3771.822
0.7760.5590.3320.2650.1880.120
0.102
0.0690.0630.051
0.0430.0370.0280.025
0.0210.021
0.0180015
0.0150.0130.013
14.2867692
3.2261.754
0.7520.5460.326
0.2620.1810.1150.101
0.0710.0580.0530.0440.0350.028
0.0260.0220.0200.0190.0160.014
0.0120.011
27.93822.55315.70911.9088.0156.8935.3594.8224.0173.216
2.9962.5362.3102.209
2.0081.7981.627
1.5801.4331.3351 3081.1961 1481.080
0.989
28.57123.07716.12912.281
8.2717.1045.5374.9874.159
3.3303.122
2.6302.3802.2722.0821.8511.6661.6121.4701.3891.3511.2501.190
1.1111.020
20.56724.91928.92030.731
32.30032.7053321833.39133.6143381133.89334.01734.02234.057
34.119
34.10434.229
34.16134.21134.10834.18134.22734.21234.20234.257
19.04823.07726.88228,65530.32630.78331.37931.58431 88732.18532.25932.43332.52132.55932.62432.70532.76932.78732.83632.86432.877329113293232.95932.990
28.44834.67940.46343.13145.62846.28647.158
47.47347.90048.34048.42948.68648.82448.863
48.92649.0784906549.15849.20349.40349.31349.33849390
49.41649.400
28.57134.61540.32342.98245.48946.17547.06847.37547.83048.2784838948.65048.78148.83848.93749.05749.153
49.18149.25449.29649.31549.367
49.39849.43849.485
7.8559.729
11.53112.40813.28013.55813.93314.04914.28114.51214.58014.69314.78314.81914.904
14.98415.05115.07715.13215.13215.17915.224
15.23415.28915.340
9.52411.53813.44114.32715.16315.39215.68915.79215.94316.09316.13016.21716.26016.279
16.31216.35216.38416.39416.41816.43216.43816.45616.46616.47916.495
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
594 PASCUAL LLÓRENTE AND JORDI QUER
Departamento de Matemática Aplicada
Edificio Matemáticas, Planta 1
Universitad de Zaragoza
50009 Zaragoza, Spain
Departament de Matemàtiques
Facultat d'Informàtica de Barcelona
Universität Politécnica de Catalunya
Pau Gargallo,5
08028 Barcelona, Spain
1. I. O. ANGELL, "A table of totally real cubic fields," Math. Comp., v. 30, 1976, pp. 184-187.2. H. DAVENPORT & H. HEILBRONN, "On the density of discriminants of cubic fields II,"
Proc. Roy. Soc. London Ser. A, v. 322, 1971, pp. 405-420.3. B. N. DELONE & D. K. FADDEEV, The Theory of Irrationalities of the Third Degree, Trudy
Mat. Inst. Steklov., vol. 11, 1940; English transi., Transi. Math. Monographs, vol. 10, Amer. Math.
Soc, Providence, R.I., Second Printing 1978.
4. V. ENNOLA k R. TURUNEN, "On totally real cubic fields," Math. Comp., v. 44, 1985,
pp. 495-518.5. H. J. GOODWIN & P. A. SAMET, "A table of real cubic fields," J. London Math. Soc, v. 34,
1959, pp. 108-110.6. P. LLÓRENTE, "Cubic irreducible polynomials in ZP[XJ and the decomposition of rational
primes in a cubic field," Publ. Sec. Mat. Univ. Autónoma Barcelona, v. 27, n. 3, 1983, pp. 5-17.
7. P. LLÓRENTE & E. NART, "Effective determination of the decomposition of the rational
primes in a cubic field," Proc. Amer. Math. Soc, v. 87, 1983, pp. 579-585.
8. P. LLÓRENTE & A. V. ONETO, "Cuerpos cúbicos," Cursos, Seminarios y Tesis del PEAM,
n. 5, Univ. Zulia, Maracaibo, Venezuela, 1980.
9. P. LLÓRENTE & A. V. ONETO, "On the real cubic fields," Math. Comp., v. 39, 1982,
pp. 689-692.10. M. SCHERING, "Théorèmes relatifs aux formes binaires quadratiques qui représentent les
mêmes nombres," Jour, de Math. (2), v. 4, 1859, pp. 253-270.
11. D. SHANKS, Review of I. O. Angell, "A table of totally real cubic fields," Math. Comp., v.
30, 1976, pp. 670-673.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use