The totally real primitive number fields of discriminant at most 10^{9}

Gunter Malle

Fachbereich Mathematik, Universität Kaiserslautern, Postfach 3049, D–67653

             Kaiserslautern, Germany. malle@mathematik.uni-kl.de

Abstract. In this note we report on the enumeration of totally real number fields of discriminant at most 10^{9} with no proper subfield and give some statistics on their properties.

1 Introduction

The enumeration of number fields with bounded discriminant (and fixed degree) has a long history. Often the aim was the determination of the field(s) with minimal possible absolute value of the discriminant for given degree and possibly given Galois group of the Galois closure. See for example the article [10]. Recently, the focus has shifted towards investigating the asymptotic behaviour of the counting function

N(G, X) := #{K/ℚ | Gal(K/ℚ) = G and |d(K/ℚ)| ≤ X}

of fields with Galois group G and discriminant bounded by X, as X goes to infinity, see for example [13, 6]. The author has put forward in [13, 14] a precise conjecture predicting the main term in the asymptotic of N(G, X), which is known to hold for abelian groups G as well as for all groups of degree at most 4 except the alternating group 𝔄_{4}.

In this note we report on the computer calculation of all primitive totally real number fields of discriminant at most 10^{9} (and in some cases even with larger discriminant). Here, a number field K/ℚ is called primitive if it contains no proper subfield except ℚ. Previously, such lists were often only computed for imprimitive fields, since for those the existence of intermediate fields provides a natural reduction to two (easier) subproblems. Moreover, imprimitive fields of degree less than 10 are solvable and thus lend themselves to methods from class field theory. Thus our results complement earlier extensive computations for imprimitive extensions.

By the unconditional Odlyzko bound [16, p.223], a totally real number field of discriminant at most 10^{9} has degree n = (K/ℚ) at most 9. Assuming the generalized Riemann hypothesis, this can be improved to n ≤ 8. By the result of Minkowski, for each fixed degree there exist only finitely many fields with bounded discriminant. Thus there is a finite number of totally real number fields of discriminant at most 10^{9}.

Theorem 1. There exist exactly 389 013 654 totally real primitive number fields of discriminant at most 10^{9}, as given in Table 1.

Table 1. Totally real primitive number fields

n number of fields
1  1
2  303,963,559
3  64,659,361
4  17,897,739
5  2,341,960
6  147,600
7  3,432
8  2
total  389,013,654

For degrees 2 and 3, as well as for degree 4 fields with Galois group 𝔄_{4}, all such fields were known before, see for example [7]. They can be found by using methods from class field theory. For degree 4 extensions with group 𝔖_{4} as well as for extensions of degree at least 5, our results are new. In particular this seems to be the first time that large numbers of degree 5 and degree 6 fields with primitive Galois group are enumerated.

The method used to find these fields is an adaptation to the totally real case of the theorem of Hunter (see [4, Thm. 6.4.2] for example) which has already often been used before. This theorem states that in order to enumerate primitive elements of all primitive fields of fixed degree and bounded discriminant, an explicitly given finite set of possible minimal polynomials has to be searched. We have implemented this algorithm in the computer algebra system KANT [9], suitably tailored to just produce totally real fields. The resulting lists of polynomials were reduced using the Pari-gp command polredabs. The total computations took several years of CPU-time on a SUN-workstation.

We use our results to study the distribution of class numbers, the maximal number of extensions with fixed discriminant, and possible asymptotic expansions of the counting functions N(G, X).

The leading term in the asymptotic behaviour of N(𝔖_{n}, X) is known for n ≤ 5, and there is a prediction by Bhargava for n ≥ 6. Our results show the following phenomenon, which had previously already been observed in degrees n = 3, 4: In the range of our tables, the actual number of 𝔖_{n}-fields is always below the proven respectively expected leading term, and consequently the first error term has a negative sign. Thus, for small discriminants, we find less 𝔖_{n}-extensions than expected. Furthermore, compared to the Cohen-Martinet heuristic for class numbers [8], we observe less fields than expected with class number larger than 1 in the range of our data. Nevertheless the proportion of such fields increases as the discriminant increases. Finally, in the range of our data the proportion of pairs, triples, ... of fields with equal discriminant increases with the size of the discriminant, so again there seem to be too few multiple discriminants at first.

2 Extensions of degree 2 and 3

The only (primitive) group in degree 2 is the cyclic group C_{2} of order 2. The number of real quadratic fields of discriminant at most 10^{9} equals 303,963,559 by [7, Table 2.2], for example.

Both groups in degree 3, the cyclic group C_{3} and the symmetric group 𝔖_{3} are primitive. The number of totally real extensions of discriminant at most 10^{9} can easily be computed using class field theory respectively the method of Belabas [1], and it equals 5008 for C_{3}-extensions respectively 64,654,353 for 𝔖_{3}-extensions by [7, Table 3.2 and 4.2].

3 Extensions of degree 4

The primitive groups in degree 4 are the alternating group 𝔄_{4} and the symmetric group 𝔖_{4}.

According to [7, Table 8.3] there are 2037 totally real 𝔄_{4}-extensions of ℚ of discriminant at most 10^{9}. It is easy to reproduce this result, obtained by using Kummer theory, with the Hunter method. The asymptotic of 𝔄_{4}-extensions of ℚ is expected to be given by

N(𝔄_{4}, X) = c_{1}X^{\frac{1}{2}} log X + c_{2}X^{\frac{1}{2}} + O(X^{α})

for positive constants c_{1}, c_{2} and some 1/4 < α < 1/2 by [7, 8.1] (see also [14] for a prediction of the leading term). This, however, has not yet been proved (see [5] for partial results).

Table 9.3 in [7] gives the number of totally real 𝔖_{4}-fields of discriminant at most 10^{7}. Computations using Hunter’s method yield the results in Table 2 for discriminant at most 10^{9}. (Note that the estimate for totally real 𝔖_{4}-fields up to 10^{8} made in [7, 9.2] is within 0.5 percent of the correct value.)

Table 2. Totally real 𝔖_{4}-fields

\begin{matrix}X & N(𝔖_{4}, X) & E(𝔖_{4}, X) & α_{1} & α_{2} & P (𝔖_{4}, X) & Δ(𝔖_{4}, X) \\ 10^{4} & 13 & 240 & 0.595 & 0.952 & 19 & 0.19 \\ 10^{5} & 449 & 2086 & 0.664 & 0.923 & 462 & 0.22 \\ 10^{6} & 8301 & 17047 & 0.705 & 0.908 & 8384 & 0.32 \\ 10^{7} & 120622 & 132855 & 0.732 & 0.892 & 120877 & 0.34 \\ 10^{8} & 1529634 & 1005137 & 0.750 & 0.876 & 1529780 & 0.27 \\ 10^{9} & 17895702 & 7464718 & 0.764 & 0.870 & 17896723 & 0.33 \end{matrix}

By the theorem of Bhargava [3] the number of totally real 𝔖_{4}-fields grows linearly with the discriminant, with proportionality constant

c_{0} := \frac{1}{48} ∏_{p≥2}(1 + \frac{1}{p^{2}} − \frac{1}{p^{3}} − \frac{1}{p^{4}} ) = 0.0253477143104 …

where the product ranges over all primes p. It seems interesting to obtain information on the error term in this asymptotic behaviour. For this assume that E(𝔖_{4}, X) := c_{0}X − N(𝔖_{4}, X) ∼ λX^{α} for some α < 1, λ > 0. Then

α_{1}(𝔖_{4}, X) := log E(𝔖_{4}, X)/ log X

and

α_{2}(𝔖_{4}, X) := log (E(𝔖_{4}, X)/E(𝔖_{4}, X/2))/ log(2)

both converge to α when X → ∞. In the fourth and fifth column of Table 2 we give α_{1}(𝔖_{4}, X) and α_{2}(𝔖_{4}, X). In the range of our data, the first increases monotonically, while the second decreases, suggesting that the exponent α of the error term should satisfy 0.76 ≤ α ≤ 0.87.

In fact, the authors of [7] expect, following a communication by Yukie, that α = 5/6, and more precisely an asymptotic behaviour of the form

N(𝔖_{4}, X) = c_{0}X + c_{1}X^{\frac{5}{6}} + c_{2}X^{\frac{3}{4}} log X + c_{3}X^{\frac{3}{4}} + O(X^{β})

with some β < 3/4. Using their count up to discriminant 10^{7} they present least squares approximations to the constants c_{1}, …, c_{3}. Assuming this form of the asymptotic expansion, using a least square method with our more extensive results we get the following approximations

c_{1} = −0.354,  c_{2} = 0.012,  c_{3} = 0.418,

to the constants appearing in the conjectured expansion for N(𝔖_{4}, X). The value of

P (𝔖_{4}, X) := c_{0}X + c_{1}X^{\frac{5}{6}} + c_{2}X^{\frac{3}{4}} log X + c_{3}X^{\frac{3}{4}} ,

with c_{1}, c_{2}, c_{3} as before, is given in column 6 of Table 2. The last column gives the quantity

Δ(𝔖_{4}, X) := log |N(𝔖_{4}, X) − P (𝔖_{4}, X)|/ log(X).

The predicted number of totally real 𝔖_{4}-fields of discriminant at most 10^{10} equals 199 133 067, which seems out of the range of the Hunter method at present.

We have used our data for 𝔖_{4} to count multiplicities of discriminants. This question has aroused some interest recently. For example, Klüners [11] has shown that the number of 𝔖_{4}-fields with discriminant D is at most equal to O_{ϵ}(D^{1/2+ϵ}) for all ϵ > 0. It is conjectured, though, that this number should rather be of the order O_{ϵ}(D^{ϵ}), an expectation consistent with our data in Table 3.

It may be remarked from the table that the number of 7-tuples of discriminants is more than twice the number of 6-tuples. Most of these 7-tuples arise

Table 3. k-tuples of 𝔖_{4}-discriminants

k  1  2  3  4 5 6 7 8 9 10 11 12 13 14 15

N(𝔖_{4})

5 · 10^{6} 4117874 191762 150463 7913 1122 388 1016 52 5 5
10^{7} 8128708 400282 319821 18453 2588 973 2485 124 19 8 1 1  1
1.5 · 10^{7} 12103274 614573 494938 29990 4288 1623 4198 197 41 21 2 3  1 1

in the following way. Take an 𝔖_{3}-field L of degree 3 whose class group contains an elementary abelian subgroup E of order 8. For any of the seven subgroups of E of order 2, there exists an unramified 𝔖_{4}-extension of L, so this gives rise to seven 𝔖_{4}-fields with the same discriminant, all containing L in their Galois closures. Similarly, one may obtain 2^{4} − 1 = 15-tuples of equal discriminants starting from 𝔖_{3}-fields with 2-rank four.

We have also calculated the class numbers of all 𝔖_{4}-fields in the range using the bnfclgp-command in Pari-gp (so the correctness of the results relies on a heuristic strengthening of the generalized Riemann hypothesis, as described in the Pari-manual). The total number of fields with given class number is displayed in Table 4.

Table 4. Class numbers of 𝔖_{4}-fields

\begin{matrix}h & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ n(h) & 15354301 & 2074924 & 187536 & 206241 & 22469 & 21858 & 5271 & 13859 & 1750 & 2362 & 722 & 1839 \end{matrix}

\begin{matrix}h & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 & 31 & 32 & 33 & 34 & 35 & 36 \\ n(h) & 322 & 541 & 176 & 683 & 103 & 143 & 51 & 166 & 36 & 70 & 17 & 84 & 16 & 27 & 13 & 35 & 2 & 9 & 2 & 17 & 6 & 4 & 6 & 8 \end{matrix}

\begin{matrix}h & 38 & 39 & 40 & 42 & 43 & 44 & 45 & 46 & 48 & 51 & 52 & 53 & 59 \\ n(h) & 6 & 3 & 5 & 2 & 4 & 2 & 1 & 2 & 3 & 1 & 2 & 1 & 1 \end{matrix}

According to the heuristic of Cohen and Martinet [8, (8.3)] the probability that the odd part h′_{K} of the class number of a totally real 𝔖_{4}-extension K/ℚ of degree 4 equals the odd integer h should be given by

pr(h′_{K} = h) = c_{4} (h^{4} ∏_{p^{j}|h}(1 − p^{−j}))^{−1}

for some explicit constant c_{4} = .978989..., where the product runs over all (odd) prime powers dividing h.

Table 5. Relative proportions of 𝔖_{4}-class numbers

h′  1  3  5  7  9  11  13
1 − 4500000 .9881 .1025E−1 .121E−2 .273E−3 .08E−3 .32E−4 .13E−4
4500001 − 9000000 .9863 .1183E−1 .142E−2 .316E−3 .09E−3 .43E−4 .22E−4
9000001 − 13500000 .9855 .1244E−1 .147E−2 .359E−3 .12E−3 .50E−4 .22E−4
13500001 − 17895702 .9852 .1274E−1 .150E−2 .360E−3 .13E−3 .53E−4 .22E−4

Cohen − Martinet .9790 .1813E−1 .196E−2 .476E−3 .25E−3 .74E−4 .37E−4

In Table 5 we give the relative proportions of the first few class numbers for chunks of 4 500 000 consecutive 𝔖_{4}-fields, as well as the proportion predicted in [8]. One observes that all class numbers bigger than 1 appear less often than expected, but the proportion slowly increases with the discriminant. In particular, the data do not seem to contradict the Cohen–Martinet-heuristic.

4 Extensions of degree 5

All transitive subgroups of 𝔖_{5} are primitive. There exist nine C_{5}-extensions of ℚ with discriminant at most 10^{9}, 302 totally real extensions with the dihedral group D_{5} and 196 totally real extensions with the Frobenius group F_{20} of order 20. The results for the alternating and the symmetric group are collected in Table 6. All these can be obtained with Hunter’s method. For square discriminants, the ideas presented in [14, Sect. 5.1] allow to get up to discriminant 10^{11}.

Table 6. Totally real 𝔄_{5}- and 𝔖_{5}-fields

\begin{matrix}X & N(𝔄_{5}, X) \\ 10^{7} & 7 \\ 10^{8} & 80 \\ 10^{9} & 398 \\ 10^{10} & 1874 \\ 10^{11} & 8121 \end{matrix}  \begin{matrix}X & N(𝔖_{5}, X) & E(𝔖_{5}, X) & α_{1} & α_{2} \\ 10^{5} & 8 & 568 & 0.551 & 0.995 \\ 10^{6} & 409 & 5348 & 0.621 & 0.970 \\ 10^{7} & 9461 & 48106 & 0.668 & 0.947 \\ 10^{8} & 162022 & 413648 & 0.702 & 0.927 \\ 10^{9} & 2341055 & 3415647 & 0.726 & 0.912 \\ 2.10^{9} & 5109739 & 6403665 & 0.732 & 0.907 \end{matrix}

By an as yet unpublished result of Bhargava (see [2]) the number of 𝔖_{5}- extensions of ℚ should grow linearly with the discriminant, with proportionality factor

c_{0} := \frac{1}{240} ∏_{p≥2}(1 + \frac{1}{p^{2}} − \frac{1}{p^{4}} − \frac{1}{p^{5}} ) = 0.005756702 …

where again p ranges over all primes. Let E(𝔖_{5}, X) := c_{0}X − N(𝔖_{5}, X). In Table 6 for 𝔖_{5} we give

α_{1}(𝔖_{5}, X) := log E(𝔖_{5}, X)/ log(X), α_{2}(𝔖_{5}, X) := log \frac{E(𝔖_{5}, X)}{E(𝔖_{5}, X/2)}/ log(2),

which seem to indicate that the exponent α of the error term in the asymptotic

N(𝔖_{5}, X) = c_{0}X + O(X^{α})

and also the position of the second right-most pole of the associated ζ-function should satisfy 0.732 ≤ α ≤ 0.907. Note, though, that in the range of our table, the error is still (slightly) larger than the main term.

Table 7. k-tuples of 𝔖_{5}-discriminants

k  1  2 3 4
N(𝔖_{5})
500000 496190 1881 16 0
1000000 991675 4114 31 1
1500000 1486819 6503 57 1
2000000 1981818 8970 78 2
2500000 2476933 11377 99 4

In Table 7 we have recorded the numbers of multiplets of 𝔖_{5}-discriminants in the range up to 10^{9}. Again the growth of the maximal multiplicity is very slow, consistent with the expectation that this maximal multiplicity should be O_{ϵ}(X^{ϵ}).

Again using the Pari-gp command bnfclgp we have calculated the class numbers of the 𝔖_{5}-fields in the range. The total number of fields with given class number is displayed in Table 8.

Table 8. Class numbers of 𝔖_{5}-fields

\begin{matrix}h & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ n(h) & 2, 292, 467 & 41, 944 & 5, 094 & 901 & 508 & 72 & 43 & 13 & 7 & 3 & 1 & 1 & 1 \end{matrix}

By the heuristic of Cohen and Martinet the probability that the part h′_{K} prime to 5 of the class number of a totally real 𝔖_{5}-extension K/ℚ equals h should be given by

pr(h′_{K} = h) = c_{5} (h^{5} ∏_{p^{j}|h}(1 − p^{−j}))^{−1}

for some explicit constant c_{5} = 0.932929..., where the product runs over all prime powers p^{j}, with p ≠ 5, dividing h. In Table 9 we give the relative proportions of the first few class numbers for chunks of 780 000 consecutive 𝔖_{5}-fields, as well as the proportion predicted in [8]. The same observations as in the 𝔖_{4}-case apply.

Table 9. Relative proportions of 𝔖_{5}-class numbers

h′  1  2  3  4  6  7  8
1 − 780000 .9837 .01433 .00166 .00024 .000018 .000009  0
780001 − 1560000 .9785 .01874 .00232 .00042 .000027 .000019 .000006
1560001 − 2340000 .9762 .02066 .00253 .00049 .000047 .000027 .000010
Cohen − Martinet .9329 .05831 .00576 .00243 .000360 .000065 .000086

5 Extensions of degree 6

The primitive groups in degree 6 are the four non-solvable groups PSL_{2}(5) ≅ 𝔄_{5}, PGL_{2}(5) ≅ 𝔖_{5}, 𝔄_{6} and 𝔖_{6}. There are precisely two PGL_{2}(5)-fields of discriminant at most 10^{9}. The data for the other three groups are collected in Table 10.

Table 10. Totally real PSL_{2}(5)-, 𝔄_{6}- and 𝔖_{6}-fields

\begin{matrix}X & N(PSL_{2}(5), X) \\ 10^{8} & 3 \\ 10^{9} & 23 \\ 10^{10} & 139 \\ 10^{11} & 720 \end{matrix}  \begin{matrix}X & N(𝔄_{6}, X) \\ 10^{9} & 22 \\ 10^{10} & 159 \end{matrix}  \begin{matrix}X & N(𝔖_{6}, X) & α_{1} \\ 10^{6} & 1 & 0.509 \\ 10^{7} & 177 & 0.578 \\ 10^{8} & 6513 & 0.629 \\ 10^{9} & 147553 & 0.666 \\ 4 · 10^{9} & 847116 & 0.684 \end{matrix}

The results for 𝔄_{6} and 𝔖_{6} were obtained by Hunter’s method, while the 𝔄_{5}- and 𝔖_{5}-fields were converted from the corresponding lists in degree 5 using a KANT-routine. According to a conjecture of Bhargava (see [2]), the number of totally real 𝔖_{6}-extensions of ℚ should grow linearly, with proportionality constant

c_{1} := \frac{1}{1440} ∏_{p≥2}(1 + \frac{1}{p^{2}} + \frac{1}{p^{3}} − \frac{2}{p^{5}} − \frac{1}{p^{6}} ) = 0.0011350200 …

In Table 10 we also give the value of α_{1}(𝔖_{6}, X), defined as in the case of 𝔖_{4} and 𝔖_{5}.

In the range considered here, there are at most two 𝔖_{6}-fields with the same discriminant. The number of pairs of fields with the same discriminant among the first N(𝔖_{6}) fields is given in Table 11.

N(𝔖_{6}) singles pairs
200000 199822  89
400000 399540 230
600000 599292 354
800000 798978 511

In Table 12 we present the number n(h) of totally teal 𝔖_{6}-fields of discriminant at most 10^{9} with class number h, again calculated with the Pari-gp command bnfclgp. It turns out that all fields have class number at most 3.

Table 12. Class numbers of 𝔖_{6}-fields

\begin{matrix}h & 1 & 2 & 3 \\ n(h) & 146, 960 & 578 & 15 \end{matrix}

6 Extensions of degree 7

All transitive subgroups of 𝔖_{7} are primitive. The totally real number fields of discriminant at most 10^{9} can be enumerated by Hunter’s method. One finds that there is one C_{7}-extension, one extension with the dihedral group D_{7}, and none with the Frobenius groups of order 21 and 42 (see also [12, Thm. 12]). There are two Kronecker-equivalent fields with group PSL_{2}(7) and one with group 𝔄_{7}. The data for the symmetric group 𝔖_{7} are given in Table 13.

Table 13. Totally real 𝔖_{7}-fields

\begin{matrix}X & N(𝔖_{7}, X) \\ 10^{8} & 71 \\ 10^{9} & 3427 \end{matrix}

According to Pari-gp, all 𝔖_{7}-fields in the range have class number 1.

7 Extensions of degree 8 and higher

There are just two totally real primitive number fields of degree 8 with discriminant at most 10^{9}, both with Galois group 𝔖_{8}, with discriminants 483,345,053 respectively 707,295,133, as can be checked by Hunter’s method (see also [12, Thm. 13]).

For degree 9, the bounds given by Hunter’s theorem are very small, and it turns out that no primitive totally real fields of discriminant at most 10^{9} exist. (According to [15] the smallest discriminant equals 9 685 993 193.) Totally real fields of degree n ≥ 10 have discriminant larger than 10^{10} by the unconditional Odlyzko bound [16, p.223].

Acknowledgement: I thank Jürgen Klüners for useful conversations on the topic of this paper.

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