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Properties

Label 20.0.67955065915...4896.6
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 41^{18}$
Root discriminant $138.55$
Ramified primes $2, 3, 41$
Class number $3904512$ (GRH)
Class group $[2, 2, 4, 4, 4, 15252]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3376968271, -201978206, 2373376453, -227128042, 778323534, -80367708, 158723365, -15048766, 22384915, -1871806, 2357049, -183046, 194491, -15754, 12613, -984, 642, -46, 25, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 25*x^18 - 46*x^17 + 642*x^16 - 984*x^15 + 12613*x^14 - 15754*x^13 + 194491*x^12 - 183046*x^11 + 2357049*x^10 - 1871806*x^9 + 22384915*x^8 - 15048766*x^7 + 158723365*x^6 - 80367708*x^5 + 778323534*x^4 - 227128042*x^3 + 2373376453*x^2 - 201978206*x + 3376968271)
 
gp: K = bnfinit(x^20 - 2*x^19 + 25*x^18 - 46*x^17 + 642*x^16 - 984*x^15 + 12613*x^14 - 15754*x^13 + 194491*x^12 - 183046*x^11 + 2357049*x^10 - 1871806*x^9 + 22384915*x^8 - 15048766*x^7 + 158723365*x^6 - 80367708*x^5 + 778323534*x^4 - 227128042*x^3 + 2373376453*x^2 - 201978206*x + 3376968271, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 25 x^{18} - 46 x^{17} + 642 x^{16} - 984 x^{15} + 12613 x^{14} - 15754 x^{13} + 194491 x^{12} - 183046 x^{11} + 2357049 x^{10} - 1871806 x^{9} + 22384915 x^{8} - 15048766 x^{7} + 158723365 x^{6} - 80367708 x^{5} + 778323534 x^{4} - 227128042 x^{3} + 2373376453 x^{2} - 201978206 x + 3376968271 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6795506591596153047335885616481274897104896=2^{30}\cdot 3^{10}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $138.55$
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:  $2, 3, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(984=2^{3}\cdot 3\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{984}(1,·)$, $\chi_{984}(797,·)$, $\chi_{984}(961,·)$, $\chi_{984}(269,·)$, $\chi_{984}(461,·)$, $\chi_{984}(529,·)$, $\chi_{984}(409,·)$, $\chi_{984}(25,·)$, $\chi_{984}(605,·)$, $\chi_{984}(865,·)$, $\chi_{984}(845,·)$, $\chi_{984}(433,·)$, $\chi_{984}(365,·)$, $\chi_{984}(221,·)$, $\chi_{984}(385,·)$, $\chi_{984}(625,·)$, $\chi_{984}(245,·)$, $\chi_{984}(769,·)$, $\chi_{984}(701,·)$, $\chi_{984}(821,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{8} + \frac{1}{3}$, $\frac{1}{657} a^{17} - \frac{2}{73} a^{16} + \frac{2}{73} a^{15} + \frac{40}{657} a^{14} + \frac{320}{657} a^{13} - \frac{106}{219} a^{12} - \frac{170}{657} a^{11} - \frac{127}{657} a^{10} - \frac{284}{657} a^{9} - \frac{308}{657} a^{8} - \frac{160}{657} a^{7} - \frac{101}{657} a^{6} + \frac{2}{73} a^{5} - \frac{193}{657} a^{4} + \frac{328}{657} a^{3} + \frac{35}{219} a^{2} + \frac{17}{219} a - \frac{272}{657}$, $\frac{1}{54531} a^{18} - \frac{40}{54531} a^{17} - \frac{2125}{18177} a^{16} + \frac{82}{54531} a^{15} + \frac{535}{54531} a^{14} - \frac{6920}{54531} a^{13} - \frac{16388}{54531} a^{12} + \frac{16534}{54531} a^{11} + \frac{14117}{54531} a^{10} + \frac{3659}{18177} a^{9} - \frac{3020}{54531} a^{8} + \frac{2981}{54531} a^{7} - \frac{4111}{54531} a^{6} - \frac{24679}{54531} a^{5} + \frac{10049}{54531} a^{4} + \frac{20702}{54531} a^{3} - \frac{680}{18177} a^{2} - \frac{16286}{54531} a + \frac{10583}{54531}$, $\frac{1}{77130717241878949211336713832545903951452407032574591087757671241} a^{19} + \frac{565183069192051225028796677397337639351985025392044663172906}{77130717241878949211336713832545903951452407032574591087757671241} a^{18} + \frac{17182211502639868978438354821701823050565721560344882129415470}{77130717241878949211336713832545903951452407032574591087757671241} a^{17} + \frac{11836024883788064307837705153487469268370691409453158202543245140}{77130717241878949211336713832545903951452407032574591087757671241} a^{16} - \frac{12218931782971727780958163912278381343365278422957512551929438328}{77130717241878949211336713832545903951452407032574591087757671241} a^{15} + \frac{3674228664795027703253992386572628231598525991768583855814303663}{25710239080626316403778904610848634650484135677524863695919223747} a^{14} - \frac{23430831353501569045526556101144249995385650276909818650876724091}{77130717241878949211336713832545903951452407032574591087757671241} a^{13} + \frac{15081806087521061353606798344760716943194408905241220030553644780}{77130717241878949211336713832545903951452407032574591087757671241} a^{12} - \frac{20733301523001078631934292206592917972735351566590730265311772117}{77130717241878949211336713832545903951452407032574591087757671241} a^{11} + \frac{28532049758776071680894409437599323175067385078278305729941506995}{77130717241878949211336713832545903951452407032574591087757671241} a^{10} - \frac{1058452744386411120962945772715719140864659360834508349703095479}{8570079693542105467926301536949544883494711892508287898639741249} a^{9} + \frac{27582340168273450973812263807802591678869278932765975212756680527}{77130717241878949211336713832545903951452407032574591087757671241} a^{8} + \frac{357907624924978201103780017106970794525971346792163075229518490}{25710239080626316403778904610848634650484135677524863695919223747} a^{7} - \frac{20349559392661333285586160261592109704507135403754967909641477332}{77130717241878949211336713832545903951452407032574591087757671241} a^{6} + \frac{5373301947207788872664823830642014201025450973021421614292229222}{77130717241878949211336713832545903951452407032574591087757671241} a^{5} + \frac{2686866884355658177524178316436799975281011479358236498537694135}{8570079693542105467926301536949544883494711892508287898639741249} a^{4} + \frac{31825859973038235672576990758433754506516710109211461832226239859}{77130717241878949211336713832545903951452407032574591087757671241} a^{3} + \frac{19296411873817067731800522661168888239411052512568780224424731808}{77130717241878949211336713832545903951452407032574591087757671241} a^{2} - \frac{22011703985448053621778478445929578624574727728643823621954682438}{77130717241878949211336713832545903951452407032574591087757671241} a + \frac{8780967958644597594446889384944329387249906945372912299952964582}{77130717241878949211336713832545903951452407032574591087757671241}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{15252}$, which has order $3904512$ (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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 5104264.636551031 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{-246}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-6}, \sqrt{41})\), 5.5.2825761.1, 10.10.327381934393961.1, 10.0.2606819247971779313664.1, 10.0.63580957267604373504.1

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 R R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.10.15.5$x^{10} + 14 x^{8} + 40 x^{6} - 144 x^{4} - 432 x^{2} + 33632$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.5$x^{10} + 14 x^{8} + 40 x^{6} - 144 x^{4} - 432 x^{2} + 33632$$2$$5$$15$$C_{10}$$[3]^{5}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41Data not computed