↑

Properties

Label 20.0.15786945900...5081.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 11^{18}\cdot 37^{10}$
Root discriminant $91.18$
Ramified primes $3, 11, 37$
Class number $1130944$ (GRH)
Class group $[4, 282736]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![44266141261, 23245166096, 12656771613, 12510065557, 15843284090, 1132845777, 6648619519, -141689449, 1379811834, -40434767, 161154278, -4273190, 11285118, -250426, 482266, -8280, 12284, -143, 171, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 171*x^18 - 143*x^17 + 12284*x^16 - 8280*x^15 + 482266*x^14 - 250426*x^13 + 11285118*x^12 - 4273190*x^11 + 161154278*x^10 - 40434767*x^9 + 1379811834*x^8 - 141689449*x^7 + 6648619519*x^6 + 1132845777*x^5 + 15843284090*x^4 + 12510065557*x^3 + 12656771613*x^2 + 23245166096*x + 44266141261)
 
gp: K = bnfinit(x^20 - x^19 + 171*x^18 - 143*x^17 + 12284*x^16 - 8280*x^15 + 482266*x^14 - 250426*x^13 + 11285118*x^12 - 4273190*x^11 + 161154278*x^10 - 40434767*x^9 + 1379811834*x^8 - 141689449*x^7 + 6648619519*x^6 + 1132845777*x^5 + 15843284090*x^4 + 12510065557*x^3 + 12656771613*x^2 + 23245166096*x + 44266141261, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 171 x^{18} - 143 x^{17} + 12284 x^{16} - 8280 x^{15} + 482266 x^{14} - 250426 x^{13} + 11285118 x^{12} - 4273190 x^{11} + 161154278 x^{10} - 40434767 x^{9} + 1379811834 x^{8} - 141689449 x^{7} + 6648619519 x^{6} + 1132845777 x^{5} + 15843284090 x^{4} + 12510065557 x^{3} + 12656771613 x^{2} + 23245166096 x + 44266141261 \)

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:  \(1578694590073830435587328344569998885081=3^{10}\cdot 11^{18}\cdot 37^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.18$
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:  $3, 11, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1221=3\cdot 11\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{1221}(1,·)$, $\chi_{1221}(260,·)$, $\chi_{1221}(73,·)$, $\chi_{1221}(1183,·)$, $\chi_{1221}(334,·)$, $\chi_{1221}(593,·)$, $\chi_{1221}(149,·)$, $\chi_{1221}(406,·)$, $\chi_{1221}(665,·)$, $\chi_{1221}(221,·)$, $\chi_{1221}(926,·)$, $\chi_{1221}(223,·)$, $\chi_{1221}(739,·)$, $\chi_{1221}(554,·)$, $\chi_{1221}(371,·)$, $\chi_{1221}(184,·)$, $\chi_{1221}(889,·)$, $\chi_{1221}(443,·)$, $\chi_{1221}(445,·)$, $\chi_{1221}(1109,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{185329} a^{11} + \frac{99}{185329} a^{9} + \frac{3564}{185329} a^{7} + \frac{56133}{185329} a^{5} - \frac{9803}{185329} a^{3} - \frac{91777}{185329} a + \frac{71773}{185329}$, $\frac{1}{185329} a^{12} + \frac{99}{185329} a^{10} + \frac{3564}{185329} a^{8} + \frac{56133}{185329} a^{6} - \frac{9803}{185329} a^{4} - \frac{91777}{185329} a^{2} + \frac{71773}{185329} a$, $\frac{1}{185329} a^{13} - \frac{6237}{185329} a^{9} + \frac{73955}{185329} a^{7} - \frac{7100}{185329} a^{5} - \frac{47925}{185329} a^{3} + \frac{71773}{185329} a^{2} + \frac{4802}{185329} a - \frac{63025}{185329}$, $\frac{1}{185329} a^{14} - \frac{6237}{185329} a^{10} + \frac{73955}{185329} a^{8} - \frac{7100}{185329} a^{6} - \frac{47925}{185329} a^{4} + \frac{71773}{185329} a^{3} + \frac{4802}{185329} a^{2} - \frac{63025}{185329} a$, $\frac{1}{1853290} a^{15} - \frac{1}{370658} a^{13} - \frac{1}{1853290} a^{12} - \frac{92714}{926645} a^{10} + \frac{83308}{926645} a^{9} - \frac{1782}{926645} a^{8} - \frac{17029}{1853290} a^{7} - \frac{797449}{1853290} a^{6} - \frac{738701}{1853290} a^{5} - \frac{845069}{1853290} a^{4} - \frac{664959}{1853290} a^{3} - \frac{330113}{1853290} a^{2} - \frac{398309}{1853290} a + \frac{764449}{1853290}$, $\frac{1}{218074750632723610} a^{16} - \frac{37355873399}{218074750632723610} a^{15} - \frac{50055485415}{43614950126544722} a^{14} + \frac{116246670357}{109037375316361805} a^{13} - \frac{519915412621}{218074750632723610} a^{12} - \frac{242741665014}{109037375316361805} a^{11} - \frac{24559518116152536}{109037375316361805} a^{10} + \frac{46303641958122456}{109037375316361805} a^{9} + \frac{78961478724309977}{218074750632723610} a^{8} + \frac{30937250481268306}{109037375316361805} a^{7} + \frac{852132000048858}{21807475063272361} a^{6} - \frac{7016332635562579}{21807475063272361} a^{5} - \frac{5034534844997224}{109037375316361805} a^{4} + \frac{21031328440585969}{109037375316361805} a^{3} + \frac{34212375127152149}{109037375316361805} a^{2} - \frac{9670307283217369}{21807475063272361} a + \frac{95845219590030609}{218074750632723610}$, $\frac{1}{218074750632723610} a^{17} + \frac{29206514327}{109037375316361805} a^{15} - \frac{568776905191}{218074750632723610} a^{14} - \frac{84296526251}{43614950126544722} a^{13} - \frac{168516820807}{218074750632723610} a^{12} + \frac{207856121508}{109037375316361805} a^{11} + \frac{51609021865261272}{109037375316361805} a^{10} + \frac{18391072827930729}{43614950126544722} a^{9} - \frac{4212514024587081}{43614950126544722} a^{8} + \frac{25572358077261239}{109037375316361805} a^{7} - \frac{5089320852921192}{21807475063272361} a^{6} - \frac{29025801189628779}{109037375316361805} a^{5} - \frac{25920512818710177}{109037375316361805} a^{4} + \frac{6969909843325798}{21807475063272361} a^{3} - \frac{31202105893120019}{109037375316361805} a^{2} - \frac{100573426839035471}{218074750632723610} a + \frac{98124775661783311}{218074750632723610}$, $\frac{1}{218074750632723610} a^{18} - \frac{452592942}{21807475063272361} a^{15} + \frac{107497754005}{43614950126544722} a^{14} + \frac{102818966966}{109037375316361805} a^{13} - \frac{75540394447}{43614950126544722} a^{12} - \frac{160330839757}{109037375316361805} a^{11} - \frac{66598481999058157}{218074750632723610} a^{10} + \frac{60742943914381577}{218074750632723610} a^{9} + \frac{8549204823353364}{21807475063272361} a^{8} + \frac{16796820409439737}{218074750632723610} a^{7} - \frac{88303784134897273}{218074750632723610} a^{6} - \frac{92813816060331659}{218074750632723610} a^{5} - \frac{89737360936246123}{218074750632723610} a^{4} + \frac{3048865231844747}{43614950126544722} a^{3} + \frac{12744593759601206}{109037375316361805} a^{2} - \frac{23037868962526627}{109037375316361805} a - \frac{96649217750661881}{218074750632723610}$, $\frac{1}{218074750632723610} a^{19} + \frac{13235719243}{218074750632723610} a^{15} + \frac{45563848091}{109037375316361805} a^{14} - \frac{58578898029}{43614950126544722} a^{13} + \frac{30055181944}{109037375316361805} a^{12} - \frac{485392046367}{218074750632723610} a^{11} + \frac{71783139337586093}{218074750632723610} a^{10} + \frac{22659046375639494}{109037375316361805} a^{9} + \frac{4966605549182969}{43614950126544722} a^{8} - \frac{13670280379055735}{43614950126544722} a^{7} + \frac{3758398115066939}{218074750632723610} a^{6} - \frac{100076611901519431}{218074750632723610} a^{5} - \frac{50555798017196597}{218074750632723610} a^{4} - \frac{9814631379637702}{21807475063272361} a^{3} - \frac{3229078132923379}{109037375316361805} a^{2} - \frac{56294818870315063}{218074750632723610} a - \frac{20183873940094979}{109037375316361805}$

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

Class group and class number

$C_{4}\times C_{282736}$, which has order $1130944$ (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:  \( 125582.779517 \) (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

\(\Q(\sqrt{33}) \), \(\Q(\sqrt{-111}) \), \(\Q(\sqrt{-407}) \), \(\Q(\sqrt{33}, \sqrt{-111})\), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{33})^+\), 10.0.3612071805471604431.1, 10.0.163509423292953287.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed
$37$37.10.5.1$x^{10} - 2738 x^{6} + 1874161 x^{2} - 11719128733$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
37.10.5.1$x^{10} - 2738 x^{6} + 1874161 x^{2} - 11719128733$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$