↑

Properties

Label 40.0.25328198843...1489.1
Degree $40$
Signature $[0, 20]$
Discriminant $11^{36}\cdot 17^{30}$
Root discriminant $72.46$
Ramified primes $11, 17$
Class number $167977$ (GRH)
Class group $[167977]$ (GRH)
Galois group $C_2\times C_{20}$ (as 40T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, 7, 12, 52, 116, 409, 1041, 3327, 9048, 27560, -44511, 108474, -195200, 472595, -762568, 2159728, -2693075, 10555266, -7000344, 56864599, 7000344, 10555266, 2693075, 2159728, 762568, 472595, 195200, 108474, 44511, 27560, -9048, 3327, -1041, 409, -116, 52, -12, 7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 + 7*x^38 - 12*x^37 + 52*x^36 - 116*x^35 + 409*x^34 - 1041*x^33 + 3327*x^32 - 9048*x^31 + 27560*x^30 + 44511*x^29 + 108474*x^28 + 195200*x^27 + 472595*x^26 + 762568*x^25 + 2159728*x^24 + 2693075*x^23 + 10555266*x^22 + 7000344*x^21 + 56864599*x^20 - 7000344*x^19 + 10555266*x^18 - 2693075*x^17 + 2159728*x^16 - 762568*x^15 + 472595*x^14 - 195200*x^13 + 108474*x^12 - 44511*x^11 + 27560*x^10 + 9048*x^9 + 3327*x^8 + 1041*x^7 + 409*x^6 + 116*x^5 + 52*x^4 + 12*x^3 + 7*x^2 + x + 1)
 
gp: K = bnfinit(x^40 - x^39 + 7*x^38 - 12*x^37 + 52*x^36 - 116*x^35 + 409*x^34 - 1041*x^33 + 3327*x^32 - 9048*x^31 + 27560*x^30 + 44511*x^29 + 108474*x^28 + 195200*x^27 + 472595*x^26 + 762568*x^25 + 2159728*x^24 + 2693075*x^23 + 10555266*x^22 + 7000344*x^21 + 56864599*x^20 - 7000344*x^19 + 10555266*x^18 - 2693075*x^17 + 2159728*x^16 - 762568*x^15 + 472595*x^14 - 195200*x^13 + 108474*x^12 - 44511*x^11 + 27560*x^10 + 9048*x^9 + 3327*x^8 + 1041*x^7 + 409*x^6 + 116*x^5 + 52*x^4 + 12*x^3 + 7*x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} + 7 x^{38} - 12 x^{37} + 52 x^{36} - 116 x^{35} + 409 x^{34} - 1041 x^{33} + 3327 x^{32} - 9048 x^{31} + 27560 x^{30} + 44511 x^{29} + 108474 x^{28} + 195200 x^{27} + 472595 x^{26} + 762568 x^{25} + 2159728 x^{24} + 2693075 x^{23} + 10555266 x^{22} + 7000344 x^{21} + 56864599 x^{20} - 7000344 x^{19} + 10555266 x^{18} - 2693075 x^{17} + 2159728 x^{16} - 762568 x^{15} + 472595 x^{14} - 195200 x^{13} + 108474 x^{12} - 44511 x^{11} + 27560 x^{10} + 9048 x^{9} + 3327 x^{8} + 1041 x^{7} + 409 x^{6} + 116 x^{5} + 52 x^{4} + 12 x^{3} + 7 x^{2} + x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 20]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(253281988439137179457025436268249370758366908512630642001927803052192601489=11^{36}\cdot 17^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.46$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(187=11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{187}(1,·)$, $\chi_{187}(4,·)$, $\chi_{187}(135,·)$, $\chi_{187}(137,·)$, $\chi_{187}(140,·)$, $\chi_{187}(13,·)$, $\chi_{187}(16,·)$, $\chi_{187}(18,·)$, $\chi_{187}(149,·)$, $\chi_{187}(152,·)$, $\chi_{187}(157,·)$, $\chi_{187}(30,·)$, $\chi_{187}(35,·)$, $\chi_{187}(38,·)$, $\chi_{187}(169,·)$, $\chi_{187}(171,·)$, $\chi_{187}(174,·)$, $\chi_{187}(47,·)$, $\chi_{187}(50,·)$, $\chi_{187}(52,·)$, $\chi_{187}(183,·)$, $\chi_{187}(186,·)$, $\chi_{187}(64,·)$, $\chi_{187}(67,·)$, $\chi_{187}(69,·)$, $\chi_{187}(72,·)$, $\chi_{187}(81,·)$, $\chi_{187}(84,·)$, $\chi_{187}(86,·)$, $\chi_{187}(89,·)$, $\chi_{187}(98,·)$, $\chi_{187}(101,·)$, $\chi_{187}(103,·)$, $\chi_{187}(106,·)$, $\chi_{187}(115,·)$, $\chi_{187}(118,·)$, $\chi_{187}(120,·)$, $\chi_{187}(123,·)$, $\chi_{187}(21,·)$, $\chi_{187}(166,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{30887} a^{22} + \frac{14665}{30887} a^{11} - \frac{1}{30887}$, $\frac{1}{30887} a^{23} + \frac{14665}{30887} a^{12} - \frac{1}{30887} a$, $\frac{1}{30887} a^{24} + \frac{14665}{30887} a^{13} - \frac{1}{30887} a^{2}$, $\frac{1}{30887} a^{25} + \frac{14665}{30887} a^{14} - \frac{1}{30887} a^{3}$, $\frac{1}{30887} a^{26} + \frac{14665}{30887} a^{15} - \frac{1}{30887} a^{4}$, $\frac{1}{30887} a^{27} + \frac{14665}{30887} a^{16} - \frac{1}{30887} a^{5}$, $\frac{1}{30887} a^{28} + \frac{14665}{30887} a^{17} - \frac{1}{30887} a^{6}$, $\frac{1}{30887} a^{29} + \frac{14665}{30887} a^{18} - \frac{1}{30887} a^{7}$, $\frac{1}{61774} a^{30} - \frac{1}{61774} a^{27} - \frac{1}{61774} a^{24} - \frac{1}{2} a^{21} - \frac{8111}{30887} a^{19} - \frac{1}{2} a^{18} + \frac{8111}{30887} a^{16} - \frac{1}{2} a^{15} + \frac{8111}{30887} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} + \frac{15443}{30887} a^{8} - \frac{1}{2} a^{6} - \frac{15443}{30887} a^{5} - \frac{1}{2} a^{3} - \frac{15443}{30887} a^{2} - \frac{1}{2}$, $\frac{1}{2502643205086} a^{31} + \frac{1864899}{1251321602543} a^{30} + \frac{18391548}{1251321602543} a^{29} - \frac{14404307}{2502643205086} a^{28} - \frac{2122327}{1251321602543} a^{27} + \frac{15948944}{1251321602543} a^{26} - \frac{27527437}{2502643205086} a^{25} - \frac{7001458}{1251321602543} a^{24} - \frac{16996693}{1251321602543} a^{23} - \frac{28423657}{2502643205086} a^{22} + \frac{3353886}{40512889} a^{21} - \frac{459436411361}{1251321602543} a^{20} + \frac{1142261837647}{2502643205086} a^{19} - \frac{258229617514}{1251321602543} a^{18} + \frac{619344037126}{1251321602543} a^{17} - \frac{835751516989}{2502643205086} a^{16} + \frac{144736154469}{1251321602543} a^{15} + \frac{71026068932}{1251321602543} a^{14} + \frac{960899431585}{2502643205086} a^{13} - \frac{148881474595}{1251321602543} a^{12} - \frac{201973606631}{1251321602543} a^{11} - \frac{18174391}{81025778} a^{10} + \frac{358778743506}{1251321602543} a^{9} + \frac{503945083650}{1251321602543} a^{8} + \frac{979623708833}{2502643205086} a^{7} - \frac{580651209231}{1251321602543} a^{6} - \frac{89154593088}{1251321602543} a^{5} + \frac{690076806417}{2502643205086} a^{4} + \frac{552173666813}{1251321602543} a^{3} - \frac{492968297838}{1251321602543} a^{2} + \frac{972395948989}{2502643205086} a + \frac{517051796822}{1251321602543}$, $\frac{1}{2502643205086} a^{32} + \frac{1}{2502643205086} a^{30} - \frac{36783091}{2502643205086} a^{29} + \frac{18391548}{1251321602543} a^{28} - \frac{14404307}{2502643205086} a^{27} + \frac{36268235}{2502643205086} a^{26} + \frac{15948944}{1251321602543} a^{25} - \frac{27527437}{2502643205086} a^{24} + \frac{26509973}{2502643205086} a^{23} - \frac{16996693}{1251321602543} a^{22} + \frac{918872944713}{2502643205086} a^{21} - \frac{33805117}{81025778} a^{20} - \frac{459436411361}{1251321602543} a^{19} - \frac{703181282081}{2502643205086} a^{18} + \frac{734862367515}{2502643205086} a^{17} + \frac{619344037126}{1251321602543} a^{16} + \frac{1009691602739}{2502643205086} a^{15} - \frac{961849293605}{2502643205086} a^{14} + \frac{71026068932}{1251321602543} a^{13} + \frac{303699346227}{2502643205086} a^{12} + \frac{953558653353}{2502643205086} a^{11} + \frac{252558257845}{1251321602543} a^{10} - \frac{18174391}{81025778} a^{9} - \frac{533764115531}{2502643205086} a^{8} - \frac{121695461177}{1251321602543} a^{7} + \frac{979623708833}{2502643205086} a^{6} + \frac{90019184081}{2502643205086} a^{5} + \frac{536485951739}{1251321602543} a^{4} + \frac{690076806417}{2502643205086} a^{3} - \frac{146974268917}{2502643205086} a^{2} + \frac{132672246989}{1251321602543} a + \frac{972395948989}{2502643205086}$, $\frac{1}{2502643205086} a^{33} - \frac{5989964}{1251321602543} a^{22} - \frac{362086439097}{1251321602543} a^{11} + \frac{1136131357197}{2502643205086}$, $\frac{1}{2502643205086} a^{34} - \frac{5989964}{1251321602543} a^{23} - \frac{362086439097}{1251321602543} a^{12} + \frac{1136131357197}{2502643205086} a$, $\frac{1}{2502643205086} a^{35} - \frac{5989964}{1251321602543} a^{24} - \frac{362086439097}{1251321602543} a^{13} + \frac{1136131357197}{2502643205086} a^{2}$, $\frac{1}{2502643205086} a^{36} - \frac{5989964}{1251321602543} a^{25} - \frac{362086439097}{1251321602543} a^{14} + \frac{1136131357197}{2502643205086} a^{3}$, $\frac{1}{2502643205086} a^{37} - \frac{5989964}{1251321602543} a^{26} - \frac{362086439097}{1251321602543} a^{15} + \frac{1136131357197}{2502643205086} a^{4}$, $\frac{1}{2502643205086} a^{38} - \frac{5989964}{1251321602543} a^{27} - \frac{362086439097}{1251321602543} a^{16} + \frac{1136131357197}{2502643205086} a^{5}$, $\frac{1}{2502643205086} a^{39} - \frac{5989964}{1251321602543} a^{28} - \frac{362086439097}{1251321602543} a^{17} + \frac{1136131357197}{2502643205086} a^{6}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{167977}$, which has order $167977$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{5185766156}{1251321602543} a^{39} + \frac{632616799455720}{1251321602543} a^{28} - \frac{1732449050492489480}{1251321602543} a^{17} - \frac{739245612967895}{1251321602543} a^{6} \) (order $22$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 65411797196858550 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{20}$ (as 40T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2\times C_{20}$
Character table for $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-187}) \), \(\Q(\sqrt{-11}, \sqrt{17})\), 4.4.4913.1, 4.0.594473.1, \(\Q(\zeta_{11})^+\), 8.0.353398147729.1, \(\Q(\zeta_{11})\), 10.10.304358957700017.1, 10.0.3347948534700187.1, 20.0.11208759391001129236841977834969.1, 20.20.131527565972137936816816034072938673.1, 20.0.15914835482628690354834740122825579433.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{4}$ $20^{2}$ $20^{2}$ $20^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
11Data not computed
17Data not computed