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Properties

Label 20.0.44774778844...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 5^{10}\cdot 71^{16}$
Root discriminant $270.74$
Ramified primes $2, 5, 71$
Class number $178545422$ (GRH)
Class group $[11, 16231402]$ (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![1874870161, 2006236288, 2387093088, 1741992792, 1262159266, 768578152, 428878820, 202581748, 83682535, 30012420, 10098714, 2555012, 23270, -261320, -65174, 4964, 3650, 112, -86, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 86*x^18 + 112*x^17 + 3650*x^16 + 4964*x^15 - 65174*x^14 - 261320*x^13 + 23270*x^12 + 2555012*x^11 + 10098714*x^10 + 30012420*x^9 + 83682535*x^8 + 202581748*x^7 + 428878820*x^6 + 768578152*x^5 + 1262159266*x^4 + 1741992792*x^3 + 2387093088*x^2 + 2006236288*x + 1874870161)
 
gp: K = bnfinit(x^20 - 4*x^19 - 86*x^18 + 112*x^17 + 3650*x^16 + 4964*x^15 - 65174*x^14 - 261320*x^13 + 23270*x^12 + 2555012*x^11 + 10098714*x^10 + 30012420*x^9 + 83682535*x^8 + 202581748*x^7 + 428878820*x^6 + 768578152*x^5 + 1262159266*x^4 + 1741992792*x^3 + 2387093088*x^2 + 2006236288*x + 1874870161, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 86 x^{18} + 112 x^{17} + 3650 x^{16} + 4964 x^{15} - 65174 x^{14} - 261320 x^{13} + 23270 x^{12} + 2555012 x^{11} + 10098714 x^{10} + 30012420 x^{9} + 83682535 x^{8} + 202581748 x^{7} + 428878820 x^{6} + 768578152 x^{5} + 1262159266 x^{4} + 1741992792 x^{3} + 2387093088 x^{2} + 2006236288 x + 1874870161 \)

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:  \(4477477884486357182574183321487582167040000000000=2^{40}\cdot 5^{10}\cdot 71^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $270.74$
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:  $2, 5, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2840=2^{3}\cdot 5\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{2840}(1,·)$, $\chi_{2840}(2439,·)$, $\chi_{2840}(1161,·)$, $\chi_{2840}(2699,·)$, $\chi_{2840}(2581,·)$, $\chi_{2840}(1421,·)$, $\chi_{2840}(1619,·)$, $\chi_{2840}(341,·)$, $\chi_{2840}(199,·)$, $\chi_{2840}(1119,·)$, $\chi_{2840}(1761,·)$, $\chi_{2840}(999,·)$, $\chi_{2840}(2539,·)$, $\chi_{2840}(1261,·)$, $\chi_{2840}(2419,·)$, $\chi_{2840}(2561,·)$, $\chi_{2840}(1141,·)$, $\chi_{2840}(2681,·)$, $\chi_{2840}(1019,·)$, $\chi_{2840}(1279,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{23} a^{12} + \frac{3}{23} a^{11} - \frac{1}{23} a^{10} + \frac{5}{23} a^{9} + \frac{5}{23} a^{8} + \frac{6}{23} a^{7} - \frac{7}{23} a^{6} - \frac{11}{23} a^{5} - \frac{10}{23} a^{3} - \frac{10}{23} a^{2} + \frac{3}{23} a - \frac{7}{23}$, $\frac{1}{943} a^{13} - \frac{7}{943} a^{12} - \frac{399}{943} a^{11} + \frac{199}{943} a^{10} - \frac{321}{943} a^{9} - \frac{113}{943} a^{8} - \frac{228}{943} a^{7} - \frac{447}{943} a^{6} - \frac{212}{943} a^{5} - \frac{332}{943} a^{4} + \frac{458}{943} a^{3} - \frac{150}{943} a^{2} + \frac{400}{943} a + \frac{139}{943}$, $\frac{1}{943} a^{14} + \frac{3}{943} a^{12} - \frac{298}{943} a^{11} - \frac{14}{41} a^{10} - \frac{105}{943} a^{9} + \frac{293}{943} a^{8} - \frac{280}{943} a^{7} + \frac{103}{943} a^{6} - \frac{176}{943} a^{5} + \frac{20}{943} a^{4} + \frac{432}{943} a^{3} - \frac{445}{943} a^{2} - \frac{423}{943} a - \frac{298}{943}$, $\frac{1}{2829} a^{15} - \frac{1}{2829} a^{13} + \frac{58}{2829} a^{12} + \frac{1315}{2829} a^{11} - \frac{1229}{2829} a^{10} + \frac{1331}{2829} a^{9} - \frac{339}{943} a^{8} + \frac{1097}{2829} a^{7} - \frac{228}{943} a^{6} - \frac{854}{2829} a^{5} - \frac{1069}{2829} a^{4} + \frac{101}{2829} a^{3} - \frac{274}{2829} a^{2} - \frac{914}{2829} a + \frac{40}{123}$, $\frac{1}{2829} a^{16} - \frac{1}{2829} a^{14} + \frac{1}{2829} a^{13} - \frac{8}{2829} a^{12} - \frac{626}{2829} a^{11} + \frac{197}{2829} a^{10} + \frac{61}{943} a^{9} - \frac{1072}{2829} a^{8} - \frac{283}{943} a^{7} - \frac{98}{2829} a^{6} - \frac{1162}{2829} a^{5} - \frac{778}{2829} a^{4} - \frac{673}{2829} a^{3} - \frac{605}{2829} a^{2} + \frac{1244}{2829} a + \frac{434}{943}$, $\frac{1}{65067} a^{17} + \frac{8}{65067} a^{16} - \frac{34}{65067} a^{14} + \frac{1}{2829} a^{13} + \frac{1210}{65067} a^{12} + \frac{21050}{65067} a^{11} + \frac{593}{65067} a^{10} - \frac{6836}{65067} a^{9} - \frac{4952}{65067} a^{8} - \frac{1768}{21689} a^{7} + \frac{20146}{65067} a^{6} + \frac{22315}{65067} a^{5} + \frac{25961}{65067} a^{4} - \frac{394}{1587} a^{3} + \frac{5861}{21689} a^{2} + \frac{23489}{65067} a + \frac{16568}{65067}$, $\frac{1}{26871231712241187779223} a^{18} + \frac{23709815348783380}{26871231712241187779223} a^{17} + \frac{404760264558932269}{26871231712241187779223} a^{16} - \frac{210477022663887136}{26871231712241187779223} a^{15} - \frac{356458923398410546}{8957077237413729259741} a^{14} + \frac{10952749844187248876}{26871231712241187779223} a^{13} + \frac{474604130541469756846}{26871231712241187779223} a^{12} + \frac{475034137818640372487}{8957077237413729259741} a^{11} + \frac{138927038141555347381}{655395895420516775103} a^{10} + \frac{3182418202301505985944}{8957077237413729259741} a^{9} + \frac{6628547864800467196466}{26871231712241187779223} a^{8} + \frac{122776769496273894425}{655395895420516775103} a^{7} + \frac{2901907307636400450146}{8957077237413729259741} a^{6} - \frac{11973016315959896119247}{26871231712241187779223} a^{5} + \frac{644434478851841334593}{26871231712241187779223} a^{4} + \frac{1434573254205490652606}{26871231712241187779223} a^{3} - \frac{6981621592428741551227}{26871231712241187779223} a^{2} - \frac{94963577499431109197}{218465298473505591701} a + \frac{12381623581866228556252}{26871231712241187779223}$, $\frac{1}{10437941033135815563078682370204940305838885129} a^{19} + \frac{71262446148564250445551}{10437941033135815563078682370204940305838885129} a^{18} + \frac{24347700089171696606535750587009672540755}{10437941033135815563078682370204940305838885129} a^{17} - \frac{352869519527845449322309340687894942043814}{10437941033135815563078682370204940305838885129} a^{16} + \frac{528686422025056931901010296692096566209564}{3479313677711938521026227456734980101946295043} a^{15} + \frac{1006411145142989331079452181040369795530889}{10437941033135815563078682370204940305838885129} a^{14} + \frac{4417069530436622636309948755619038227413284}{10437941033135815563078682370204940305838885129} a^{13} + \frac{55568549219231021815390363054694235934652551}{3479313677711938521026227456734980101946295043} a^{12} - \frac{1840994902708837005201166692541528988180222095}{10437941033135815563078682370204940305838885129} a^{11} + \frac{103559302433417548311686477078266866010907823}{3479313677711938521026227456734980101946295043} a^{10} + \frac{55254135049218859741531521531099619964993830}{254583927637458916172650789517193665996070369} a^{9} - \frac{3542251893489954086292566194373246471346083015}{10437941033135815563078682370204940305838885129} a^{8} + \frac{929188148813713851310329253623141500791341441}{3479313677711938521026227456734980101946295043} a^{7} - \frac{2203071046595661827465592076582512401122877789}{10437941033135815563078682370204940305838885129} a^{6} - \frac{3555490645033389972474478706892776094197511682}{10437941033135815563078682370204940305838885129} a^{5} + \frac{3999424816107369778782193289459690108009314622}{10437941033135815563078682370204940305838885129} a^{4} - \frac{2198982149710195926788210659105827388205867182}{10437941033135815563078682370204940305838885129} a^{3} - \frac{1719672957320389791092644722850389933054448892}{3479313677711938521026227456734980101946295043} a^{2} + \frac{2281979447874759109362563711934706900682009431}{10437941033135815563078682370204940305838885129} a - \frac{1648733696362573607977605267614274944356770117}{3479313677711938521026227456734980101946295043}$

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

Class group and class number

$C_{11}\times C_{16231402}$, which has order $178545422$ (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:  \( 116573225.49574807 \) (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

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{-5})\), 5.5.25411681.1, 10.0.2066411299986435200000.1, 10.0.66125161599565926400000.1, 10.10.21160051711861096448.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
5Data not computed
$71$71.10.8.1$x^{10} - 39121 x^{5} + 811858091$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
71.10.8.1$x^{10} - 39121 x^{5} + 811858091$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$