Properties

Label 20.0.14308691778...6464.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{10}\cdot 31^{3}\cdot 2617^{4}$
Root discriminant $11.42$
Ramified primes $2, 31, 2617$
Class number $1$
Class group Trivial
Galois Group 20T1015

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 5, -7, 17, -5, -9, 40, -22, -2, 27, -20, 5, 11, -20, 16, -1, -11, 11, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 11*x^18 - 11*x^17 - x^16 + 16*x^15 - 20*x^14 + 11*x^13 + 5*x^12 - 20*x^11 + 27*x^10 - 2*x^9 - 22*x^8 + 40*x^7 - 9*x^6 - 5*x^5 + 17*x^4 - 7*x^3 + 5*x^2 - 3*x + 1)
gp: K = bnfinit(x^20 - 5*x^19 + 11*x^18 - 11*x^17 - x^16 + 16*x^15 - 20*x^14 + 11*x^13 + 5*x^12 - 20*x^11 + 27*x^10 - 2*x^9 - 22*x^8 + 40*x^7 - 9*x^6 - 5*x^5 + 17*x^4 - 7*x^3 + 5*x^2 - 3*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 5 x^{19} \) \(\mathstrut +\mathstrut 11 x^{18} \) \(\mathstrut -\mathstrut 11 x^{17} \) \(\mathstrut -\mathstrut x^{16} \) \(\mathstrut +\mathstrut 16 x^{15} \) \(\mathstrut -\mathstrut 20 x^{14} \) \(\mathstrut +\mathstrut 11 x^{13} \) \(\mathstrut +\mathstrut 5 x^{12} \) \(\mathstrut -\mathstrut 20 x^{11} \) \(\mathstrut +\mathstrut 27 x^{10} \) \(\mathstrut -\mathstrut 2 x^{9} \) \(\mathstrut -\mathstrut 22 x^{8} \) \(\mathstrut +\mathstrut 40 x^{7} \) \(\mathstrut -\mathstrut 9 x^{6} \) \(\mathstrut -\mathstrut 5 x^{5} \) \(\mathstrut +\mathstrut 17 x^{4} \) \(\mathstrut -\mathstrut 7 x^{3} \) \(\mathstrut +\mathstrut 5 x^{2} \) \(\mathstrut -\mathstrut 3 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(1430869177844446526464=2^{10}\cdot 31^{3}\cdot 2617^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.42$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 31, 2617$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{272085514565} a^{19} - \frac{92853578442}{272085514565} a^{18} + \frac{15888810470}{54417102913} a^{17} + \frac{69553206389}{272085514565} a^{16} + \frac{65420503271}{272085514565} a^{15} + \frac{7720682619}{272085514565} a^{14} - \frac{44089421458}{272085514565} a^{13} + \frac{88201761702}{272085514565} a^{12} + \frac{78281952096}{272085514565} a^{11} + \frac{40337167528}{272085514565} a^{10} + \frac{117992989721}{272085514565} a^{9} - \frac{128966888769}{272085514565} a^{8} + \frac{81784740281}{272085514565} a^{7} - \frac{12603299272}{272085514565} a^{6} - \frac{26314477145}{54417102913} a^{5} + \frac{21533092220}{54417102913} a^{4} - \frac{127958118368}{272085514565} a^{3} + \frac{103342545059}{272085514565} a^{2} - \frac{23029136343}{272085514565} a - \frac{99283461742}{272085514565}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( \frac{60133888873}{272085514565} a^{19} - \frac{298778534621}{272085514565} a^{18} + \frac{125749937816}{54417102913} a^{17} - \frac{565943696058}{272085514565} a^{16} - \frac{154497353552}{272085514565} a^{15} + \frac{880885769627}{272085514565} a^{14} - \frac{902910373719}{272085514565} a^{13} + \frac{330553090031}{272085514565} a^{12} + \frac{537516957093}{272085514565} a^{11} - \frac{1245219308116}{272085514565} a^{10} + \frac{1311304839963}{272085514565} a^{9} + \frac{291306858373}{272085514565} a^{8} - \frac{1453679251817}{272085514565} a^{7} + \frac{1815058065489}{272085514565} a^{6} - \frac{53766486376}{54417102913} a^{5} - \frac{188743143152}{54417102913} a^{4} + \frac{771328825651}{272085514565} a^{3} - \frac{362791202978}{272085514565} a^{2} - \frac{395377999964}{272085514565} a - \frac{71740030276}{272085514565} \),  \( \frac{73550967561}{272085514565} a^{19} - \frac{358776658262}{272085514565} a^{18} + \frac{142299674631}{54417102913} a^{17} - \frac{436406983611}{272085514565} a^{16} - \frac{744471731989}{272085514565} a^{15} + \frac{1616685326159}{272085514565} a^{14} - \frac{1020982344778}{272085514565} a^{13} - \frac{252445519393}{272085514565} a^{12} + \frac{1082197170616}{272085514565} a^{11} - \frac{1500089405487}{272085514565} a^{10} + \frac{1410734894131}{272085514565} a^{9} + \frac{1042918876741}{272085514565} a^{8} - \frac{2656861552999}{272085514565} a^{7} + \frac{2221579545398}{272085514565} a^{6} + \frac{272466210976}{54417102913} a^{5} - \frac{311271020603}{54417102913} a^{4} + \frac{933714806707}{272085514565} a^{3} + \frac{450012416949}{272085514565} a^{2} + \frac{254296271707}{272085514565} a - \frac{173184072872}{272085514565} \),  \( \frac{127984771237}{272085514565} a^{19} - \frac{532733281509}{272085514565} a^{18} + \frac{193034343081}{54417102913} a^{17} - \frac{707180442757}{272085514565} a^{16} - \frac{269254360793}{272085514565} a^{15} + \frac{983362978913}{272085514565} a^{14} - \frac{1092387527761}{272085514565} a^{13} + \frac{697932733944}{272085514565} a^{12} + \frac{472860699732}{272085514565} a^{11} - \frac{1556997933119}{272085514565} a^{10} + \frac{1885044705792}{272085514565} a^{9} + \frac{911584962807}{272085514565} a^{8} - \frac{783999526338}{272085514565} a^{7} + \frac{3024738593601}{272085514565} a^{6} + \frac{133067493646}{54417102913} a^{5} + \frac{232176916716}{54417102913} a^{4} + \frac{1645466705109}{272085514565} a^{3} + \frac{110315921928}{272085514565} a^{2} + \frac{517400587054}{272085514565} a - \frac{42132770904}{272085514565} \),  \( \frac{116662206199}{272085514565} a^{19} - \frac{487288654348}{272085514565} a^{18} + \frac{165299068902}{54417102913} a^{17} - \frac{321741318959}{272085514565} a^{16} - \frac{1032002613231}{272085514565} a^{15} + \frac{1773593005816}{272085514565} a^{14} - \frac{1108933642372}{272085514565} a^{13} - \frac{147476332167}{272085514565} a^{12} + \frac{1228052637539}{272085514565} a^{11} - \frac{1687633199978}{272085514565} a^{10} + \frac{1470718218004}{272085514565} a^{9} + \frac{1831636664369}{272085514565} a^{8} - \frac{2309326354126}{272085514565} a^{7} + \frac{2863526312377}{272085514565} a^{6} + \frac{444385326395}{54417102913} a^{5} - \frac{143037990508}{54417102913} a^{4} + \frac{1568461147963}{272085514565} a^{3} + \frac{482651736191}{272085514565} a^{2} + \frac{329093784468}{272085514565} a - \frac{89249173098}{272085514565} \),  \( a \),  \( \frac{121043780821}{272085514565} a^{19} - \frac{511287704607}{272085514565} a^{18} + \frac{184359513833}{54417102913} a^{17} - \frac{590104834066}{272085514565} a^{16} - \frac{576575495099}{272085514565} a^{15} + \frac{1431971428944}{272085514565} a^{14} - \frac{1265065125243}{272085514565} a^{13} + \frac{302517690527}{272085514565} a^{12} + \frac{1068191710581}{272085514565} a^{11} - \frac{1848794657447}{272085514565} a^{10} + \frac{1779139461261}{272085514565} a^{9} + \frac{1253660237991}{272085514565} a^{8} - \frac{1710299971159}{272085514565} a^{7} + \frac{3351618338858}{272085514565} a^{6} + \frac{257732785208}{54417102913} a^{5} - \frac{61669399591}{54417102913} a^{4} + \frac{1803374116507}{272085514565} a^{3} + \frac{350551941274}{272085514565} a^{2} + \frac{214119567437}{272085514565} a - \frac{78800251327}{272085514565} \),  \( \frac{26690513153}{272085514565} a^{19} - \frac{57891189996}{272085514565} a^{18} - \frac{8758705072}{54417102913} a^{17} + \frac{396983275937}{272085514565} a^{16} - \frac{738211666602}{272085514565} a^{15} + \frac{640294231932}{272085514565} a^{14} - \frac{137582305099}{272085514565} a^{13} - \frac{463344785969}{272085514565} a^{12} + \frac{779464091073}{272085514565} a^{11} - \frac{440618909511}{272085514565} a^{10} - \frac{142025019307}{272085514565} a^{9} + \frac{1278257683113}{272085514565} a^{8} - \frac{600593342562}{272085514565} a^{7} + \frac{774984880204}{272085514565} a^{6} + \frac{350800358432}{54417102913} a^{5} - \frac{65585334078}{54417102913} a^{4} + \frac{1186031476246}{272085514565} a^{3} + \frac{538173228647}{272085514565} a^{2} + \frac{253466725256}{272085514565} a - \frac{52034880576}{272085514565} \),  \( \frac{8626170245}{54417102913} a^{19} - \frac{38281173852}{54417102913} a^{18} + \frac{72495427401}{54417102913} a^{17} - \frac{51763341159}{54417102913} a^{16} - \frac{40385565630}{54417102913} a^{15} + \frac{119512886590}{54417102913} a^{14} - \frac{123743377065}{54417102913} a^{13} + \frac{72331623399}{54417102913} a^{12} + \frac{23242877453}{54417102913} a^{11} - \frac{127694611450}{54417102913} a^{10} + \frac{189634705473}{54417102913} a^{9} + \frac{13392752130}{54417102913} a^{8} - \frac{126382811347}{54417102913} a^{7} + \frac{262934825390}{54417102913} a^{6} - \frac{3479156481}{54417102913} a^{5} + \frac{50243519052}{54417102913} a^{4} + \frac{71244877364}{54417102913} a^{3} - \frac{74663797617}{54417102913} a^{2} + \frac{163365589771}{54417102913} a - \frac{31640793947}{54417102913} \),  \( \frac{279239715169}{272085514565} a^{19} - \frac{1352613605393}{272085514565} a^{18} + \frac{561297320065}{54417102913} a^{17} - \frac{2396198118229}{272085514565} a^{16} - \frac{1045895317166}{272085514565} a^{15} + \frac{4395077301911}{272085514565} a^{14} - \frac{4286021689687}{272085514565} a^{13} + \frac{1557249414303}{272085514565} a^{12} + \frac{1962963697954}{272085514565} a^{11} - \frac{5058150276318}{272085514565} a^{10} + \frac{6143952111429}{272085514565} a^{9} + \frac{1301238085734}{272085514565} a^{8} - \frac{6570577392841}{272085514565} a^{7} + \frac{9029785200267}{272085514565} a^{6} + \frac{67101723915}{54417102913} a^{5} - \frac{427626138192}{54417102913} a^{4} + \frac{3454698653708}{272085514565} a^{3} - \frac{643793005269}{272085514565} a^{2} + \frac{1119472424748}{272085514565} a - \frac{636043584023}{272085514565} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 130.791344738 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

20T1015:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 3932160
The 506 conjugacy class representatives for t20n1015 are not computed
Character table for t20n1015 is not computed

Intermediate fields

5.1.2617.1, 10.0.212309359.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ $20$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ R $20$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.10.10.14$x^{10} + 5 x^{8} - 50 x^{6} - 58 x^{4} + 49 x^{2} + 21$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
31Data not computed
2617Data not computed