Minimal Discriminants

Results known to date. One goal of our database is to provide fields with small (absolute value of the) discriminant for each Galois group and signature. In small degrees it is even possible to determine the field(s) with smallest discriminant. Let's comment on the present state of knowledge in this area.

 

It is very easy to enumerate the discriminants of quadratic fields.

 

Degree 3:

Belabas [Bel] gives a very efficient algorithm to enumerate cubic number fields.

 

Degree 4:

In [BFP] all quartic fields with absolute discriminant smaller than 106 are enumerated.

 

Degree 5:

There are huge tables of the smallest quintic fields due to [SPD]. These tables are sufficient to extract the smallest discriminants for all Galois groups and classes of involutions for degree 5.

 

Degree 6:

The minimal discriminants for all signatures of degree 6 are computed in [Poh2]. [Oli2, FoPo1, FoPo2, FPD] have finished the computation of minimal discriminants of all signatures and all primitive Galois groups of degree 6. [BMO, Oli1] compute the minimal fields for imprimitive groups of degree 6. This yields enough information to determine the minimal fields for all groups and all conjugacy classes of that degree.

  • Imprimitive fields: [BMO, Oli1]
  • Primitive fields: [Oli2, FoPo1, FoPo2, FPD, Poh2]

 

Degree 7:

In degree 7 the minimal fields of each signature are known due to [Dia1, Dia2, Poh1].

This covers all signatures of the symmetric groups.

  • Symmetric group: [Dia1, Dia2, Poh1]
  • Other non-solvable groups: [KlMa (Theorem 12, Geometry of numbers)]
  • Solvable groups: [KlMa (Theorem 12, Methods from class field theory), FiKl]

 

Degree 8:

For imprimitive octic fields with a quartic subfield [CDO] compute huge tables using class field theory which cover all imprimitive groups and all possible signatures such that the corresponding field has a quartic subfield. These tables are not sufficient to find all minimal fields of that shape such that complex conjugation lies in a given class of involutions.

 

In [FiKl] the minima for octic fields having a quadratic subfield are computed. Altogether, we cover all possible signatures for all solvable groups.

  • Octic fields with a quartic subfield: [CDO]
  • Octic fields with a quadratic subfield: [FiKl]
  • S8 totally real: [KlMa]
  • 8T25, 8T36: [FiKl]

 

The minimal discriminants for the symmetric group have been found in [Dia3, Bat1, Bat2]. We remark that in [Mal] we find a list of all totally real primitive number fields of discriminant up to 10^9 which is sufficient to prove the totally real S_n case in degree 8. The two other minima in the table below are proved in [JoRo].

 

In the following table we list the groups with at least one unproven minimal discriminant in degree 8. Here, we only know two minima, which are marked in green. The black discriminants are the smallest ones known to us.

 

Degree

Name

0

2

4

6

8

8T37 

L(8)=PSL(2,7)

37822859361

-

-

-

134579395898896

8T43 

L(8):2=PGL(2,7)

418195493

1997331875

-

-

20418048738368

8T48 

E(8):L7=AL(8)

32684089

-

293471161

-

81366421504

8T49 

A(8)

16410601

-

144889369

-

46664208361

 

(*) = the green values are proven

(*) = the black values are open

 

Degree 9:

In degree 9 there are some partial results for imprimitive fields [DiOl]. Furthermore some totally real minimal discriminants are determined [Tak]. In a recent work, [JON] finished the computation of all minima for all solvable groups.

 

The known minima for the symmetric group have been proved in [Bat2,Tak]. 

 

In the following table we list the groups with at least one unproven minimal discriminant in degree 9. The black discriminants are the smallest ones known to us. The green values are already proven.

 

Degree

Name

1

3

5

7

9

9T27

L(9)=PSL(2,8)

79082438656

-

-

-

113186808198805...
 31 digits

9T32

L(9):3=P|L(2,8)

22663495936

-

-

-

628373976482826...
 30 digits

9T33 

A(9)

92371321

-

3200504329

-

11729467378561

9T34 

S(9)

29510281

109880167

453771377

1904081383

9685993193

 

(*) = the green values are proven

(*) = the black values are open

 

Degree 10:

For many imprimitive groups the minimal discriminants are determined in [DrJo]. In the meantime these computations have been extended to all fields with absolute discriminant smaller than 1.2*1011

 

Note that the groups with missing minima are either primitive or only admit a block of size 5 which corresponds to a quadratic subfield.

 

In the following table we list the groups with at least one unproven minimal discriminant in degree 10. The black discriminants are the smallest ones known to us. The green values are already proven.

 

Degree

Name

0

2

4

6

8

10

10T6

[5^2]2

224415603

-

-

-

-

6194123253125

10T9

[1/2.D(5)^2]2

217799879264163

104256800000

-

-

-

106599209867706368

10T10

1/2[D(5)^2]2

-

292820000000

-

-

-

33422341377520625

10T17

[5^2:4]2

2626093750000

1229312000000

-

-

-

43789058048000000

10T18

[5^2:4]2_2

-

9040581643536

-

-

-

6105273987852349696

10T19

[5^2:4_2]2

1966899200000

3768595808256

-

-

-

220292710400000000

10T20

[5^2:4_2]2_2

-

234925181632512

-

-

-

114661785600000000

10T21

[D(5)^2]2

593019904

1128753125

-

34424253125

-

3256446753125

10T26

L(10)=PSL(2,9)

-

11284439629824

-

-

-

304061824840300...
 20 digits

10T27

[1/2.F(5)^2]2

67162921875

2242969600000

-

15680000000000

-

728703488000000

10T28

1/2[F(5)^2]2

-

640000000000

-

45562500000000

-

3399254384765625

10T30

L(10):2=PGL(2,9)

35664401793024

55267035185152

-

-

-

314547523847251...
 30 digits

10T31

M(10)=L(10)'2

-

268435456000000

-

-

-

856766180266359...
 23 digits

10T32

S_6(10)=L(10):2

-

95820414976

1508214295232

-

-

525501674708224

10T33

[F(5)^2]2

9300278979

47280848896

-

9932496465625

-

277597456433152

10T35

L(10).2^2=P|L(2,9)

662747776000

24207794634752

163840000000000

-

-

397310382895823...
 20 digits

10T40

[A(5)^2]2

5841576387

6652128125

-

1919520800000

-

51717300078125

10T41

[1/2.S(5)^2]2=[A(5):2]2

4859704512

12462528125

-

194242050000

-

109268775200000

10T42

1/2[S(5)^2]2

-

11103890625

-

85629390625

-

207699287474176

10T43

[S(5)^2]2

216670707

875003125

3405971875

14339628125

85992371875

911025153125

10T44

A(10)

-

1844444809

-

51471358129

-

1464365125816576

10T45

S(10)

236438047

802448461

3316535227

19388527573

94822656283

513087549389

 

(*) = the green values are proven

(*) = the black values are open

 

Degree 11:

In the following table we list the groups with at least one unproven minimal discriminant in degree 11. The black discriminants are the smallest ones known to us.

Note that all of these missing groups are not solvable. The minima of the solvable groups in degree 11 have been found with the methods described in [FiKl].

 

Degree

Name

1

3

5

7

9

11

11T5

L(11)=PSL(2,11)(11)

-

11239665258721

-

-

-

313765182433896...
 20 digits

11T6

M(11)

-

95241470237660224

-

-

-

118769262421915...
 45 digits

11T7

A(11)

-

64283038681

-

47353198025956

-

952203981320302...
 32 digits

11T8

S(11)

5939843699

24963663301

132326332471

610429790897

7530807227563

48706494267293

 

(*) = the black values are open

 

Degree 13:

In the following table we list the groups with at least one unproven minimal discriminant in degree 13. The black discriminants are the smallest ones known to us.

Note that all of these missing groups are not solvable. The minima of the solvable groups in degree 13 have been found with the methods described in [FiKl].

 

Degree

Name

1

3

5

7

9

11

13

13T7

L(13)=PSL(3,3)

-

-

8423789045905096704

-

-

-

362838554526023...
 33 digits

13T8

A(13)

57655000561921

-

5749519947196921

-

810753495582814...
 59 digits

-

308767218127767...
 44 digits

13T9

S(13)

1325925503633

5570916369223

28261626739249

137400291790087

737652920184769

4161299413431551

50359924122392641

 

(*) = the black values are open

 

 

In order to see the statistics for other degrees, please click on Statistics and then on "Groups with unproven minimal discriminants".

 

 

References:

 

  • [Bat1] F. Battistoni,The minimum discriminant of number fields of degree 8 and signature (2,3), J. Number Theory 198, 386-395 (2019).
  • [Bat2] F. Battistoni,On small discriminants of number fields of degree 8 and 9.
  • [Bel] K. Belabas, A fast algorithm to compute cubic fields, Math. Comput. 66, No.219, 1213-1237 (1997).
  • [BFP] J. Buchmann, D. Ford, and M. Pohst, Enumeration of quartic fields of small discriminant, Math. Comput. 61, No.204, 873-879 (1993).
  • [BMO] A. Berge, J. Martinet, and M. Olivier, The computation of sextic fields with a quadratic subfield, Math. Comput. 54, No.190, 869-884 (1990).
  • [CDO] H. Cohen; F. Diaz y Diaz; M. Olivier, Tables of octic fields with a quartic subfield, Math. Comp. 68, No. 228, 1701-1716 (1999).
  • [Dia1] F. Diaz y Diaz, Discriminant minimal et petits discriminants des corps de nombres de degree 7 avec cinq places reelles, J. London Math. Soc., II. Ser. 38, 33-46 (1988).
  • [Dia2] F. Diaz y Diaz, Valeurs minima du discriminant pour certains types de corps de degree 7, Ann. Inst. Fourier 34, No.3, 29-38 (1984).
  • [Dia3] F. Diaz y Diaz, Petits discriminants des corps de nombres totalement imaginaires de degree 8, J.Numb.Th., 25, 34-52 (1987).
  • [DiOl] F. Diaz y Diaz, M. Olivier, Imprimitive ninth-degree number fields with small discriminants. With microfiche supplement. Math. Comp. 64, no. 209, 305-321 (1995).
  • [DrJo] E. Driver, J. Jones, Minimum discriminants of imprimitive decic fields. Experiment. Math. 19, No. 4, 475-479 (2010).
  • [FiKl] C. Fieker and J. Klüners, Minimal Discriminants for Fields with Frobenius Groups as Galois Groups, J.Numb.Th., 99, 318-337 (2003).
  • [FoPo1] D. Ford and M. Pohst, The totally real A5 extension of degree 6 with minimum discriminant, Exp. Math. 1, No.3, 231-235 (1992).
  • [FoPo2] D. Ford and M. Pohst, The totally real A6 extension of degree 6 with minimum discriminant, Exp. Math. 2, No.3, 231-232 (1993).
  • [FPD] D. Ford, M. Pohst, M. Daberkow, and H. Nasser, The S5 extensions of degree 6 with minimum discriminant, Exp. Math. 7, No.2, 121-124 (1998).
  • [IDPF] J. Carmelo Interlando, J. O. Dantas Lopes, T. Pires da Nobrega Neto, A.L. Flores; On the minimum absolute value of the discriminant of abelian fields of degree p2. J. Algebra Appl. 9, no. 5, 819-824 (2010).
  • [JON] J. Jones, Minimal solvable nonic fields, LMS J. Comput. Math., 16, 130–138 (2013).
  • [JoRo] John W. Jones, David P. Roberts, Mixed degree number field computations, Ramanujan J. 47, No. 1, 47–66 (2018).
  • [KlMa] J. Klüners and G. Malle, A Database for Field Extensions of the Rationals, LMS J. Comput. Math., 4, 182-196 (2001).
  • [Let] P. Letard, Valeur minimum des discriminants des corps de nombres de degré 9 totalement réels sous GRH (hypothèse de Riemann généralisée), C. R. Acad. Sci. Paris Sér. I Math. 320, no. 2, 135–138 (1995).
  • [Ma] G. Malle, The totally real primitive number fields of discriminant at most 10^9, Algorithmic number theory, 114–123 (2006).
  • [Oli1] M. Olivier, The computation of sextic fields with a cubic subfield and no quadratic subfield, Math. Comput. 58, No.197, 419-432 (1992).
  • [Oli2] M. Olivier, Corps sextiques primitifs. IV, Semin. Theor. Nombres Bordx., Ser. II 3, No.2, 381-404 (1991).
  • [Poh1] M. Pohst, The minimum discriminant of seventh degree totally real algebraic number fields, Number Theory and Algebra; Collect. Pap. dedic. H. B. Mann, A. E. Ross, O. Taussky-Todd, 235-240 (1977).
  • [Poh2] M. Pohst, On the computation of number fields of small discriminants including the minimum discriminants of sixth degree fields, J. Number Theory 14, 99-117 (1982).
  • [SPD] A. Schwarz, M. Pohst, and F. Diaz y Diaz, A table of quintic number fields, Math. Comput. 63, No.207, 361-376 (1994).
  • [Tak] K. Takeuchi, Totally real algebraic number fields of degree 9 with small discriminant, Saitama Math. J. 17, 63-85 (1999).

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