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Properties

Label 20.0.32225855565...3136.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 67^{2}\cdot 401^{8}$
Root discriminant $47.36$
Ramified primes $2, 67, 401$
Class number $39$ (GRH)
Class group $[39]$ (GRH)
Galois group 20T141

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![145247, -150018, 16835, 33158, 17985, 22698, 33296, -42100, 9710, -23534, 20357, 1344, -6037, 1598, 390, -222, 15, -16, 22, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 22*x^18 - 16*x^17 + 15*x^16 - 222*x^15 + 390*x^14 + 1598*x^13 - 6037*x^12 + 1344*x^11 + 20357*x^10 - 23534*x^9 + 9710*x^8 - 42100*x^7 + 33296*x^6 + 22698*x^5 + 17985*x^4 + 33158*x^3 + 16835*x^2 - 150018*x + 145247)
 
gp: K = bnfinit(x^20 - 8*x^19 + 22*x^18 - 16*x^17 + 15*x^16 - 222*x^15 + 390*x^14 + 1598*x^13 - 6037*x^12 + 1344*x^11 + 20357*x^10 - 23534*x^9 + 9710*x^8 - 42100*x^7 + 33296*x^6 + 22698*x^5 + 17985*x^4 + 33158*x^3 + 16835*x^2 - 150018*x + 145247, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 22 x^{18} - 16 x^{17} + 15 x^{16} - 222 x^{15} + 390 x^{14} + 1598 x^{13} - 6037 x^{12} + 1344 x^{11} + 20357 x^{10} - 23534 x^{9} + 9710 x^{8} - 42100 x^{7} + 33296 x^{6} + 22698 x^{5} + 17985 x^{4} + 33158 x^{3} + 16835 x^{2} - 150018 x + 145247 \)

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:  \(3222585556571212807579035957723136=2^{30}\cdot 67^{2}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.36$
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:  $2, 67, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{14} - \frac{2}{9} a^{12} + \frac{4}{9} a^{11} + \frac{4}{9} a^{9} + \frac{4}{9} a^{8} + \frac{1}{3} a^{7} - \frac{2}{9} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} - \frac{4}{9} a^{2} + \frac{4}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{15} + \frac{1}{9} a^{13} + \frac{4}{9} a^{12} + \frac{1}{9} a^{10} - \frac{2}{9} a^{9} - \frac{2}{9} a^{7} + \frac{1}{9} a^{4} - \frac{1}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{755217} a^{18} - \frac{959}{27971} a^{17} + \frac{30218}{755217} a^{16} + \frac{28741}{251739} a^{15} - \frac{110740}{755217} a^{14} + \frac{52639}{755217} a^{13} + \frac{175759}{755217} a^{12} - \frac{368638}{755217} a^{11} - \frac{171620}{755217} a^{10} + \frac{35776}{755217} a^{9} + \frac{304817}{755217} a^{8} + \frac{27428}{83913} a^{7} + \frac{96787}{755217} a^{6} - \frac{150806}{755217} a^{5} - \frac{100471}{755217} a^{4} + \frac{78608}{755217} a^{3} - \frac{101846}{251739} a^{2} - \frac{255941}{755217} a - \frac{518}{2241}$, $\frac{1}{1521111298836538707538234019407502832755547971337} a^{19} + \frac{68715351735626004259111932735165485827958}{169012366537393189726470446600833648083949774593} a^{18} + \frac{66883146816375040861072249104925567416442112327}{1521111298836538707538234019407502832755547971337} a^{17} + \frac{17852105251208502467989113400794231237876819359}{507037099612179569179411339802500944251849323779} a^{16} + \frac{16347869392750626578831787923305262979069336257}{1521111298836538707538234019407502832755547971337} a^{15} + \frac{130750975372736788487057985971601174444203543110}{1521111298836538707538234019407502832755547971337} a^{14} - \frac{234285789977244539661377428924852243719348660518}{1521111298836538707538234019407502832755547971337} a^{13} - \frac{296259514862122454297141451696999136744577942554}{1521111298836538707538234019407502832755547971337} a^{12} + \frac{357072604555891549459334998601551726946735288887}{1521111298836538707538234019407502832755547971337} a^{11} - \frac{98051303228838972900035307984766720668702265652}{1521111298836538707538234019407502832755547971337} a^{10} + \frac{358562340612213340571282211356874915992087789171}{1521111298836538707538234019407502832755547971337} a^{9} - \frac{71936020071626694919371612359084185647524549300}{507037099612179569179411339802500944251849323779} a^{8} + \frac{347841033046953241175650110055753674954779722261}{1521111298836538707538234019407502832755547971337} a^{7} + \frac{47670091353261558244084950115574391752386425886}{1521111298836538707538234019407502832755547971337} a^{6} + \frac{658744270961504840763564051742797396415149491651}{1521111298836538707538234019407502832755547971337} a^{5} - \frac{50695202774711433207863383940464704143326067977}{1521111298836538707538234019407502832755547971337} a^{4} - \frac{25646841360016375569159779586142499820984961475}{56337455512464396575490148866944549361316591531} a^{3} - \frac{611497650309477751508683928293531627069338073112}{1521111298836538707538234019407502832755547971337} a^{2} + \frac{497887950912879707334030466271068246304106167707}{1521111298836538707538234019407502832755547971337} a + \frac{184421911737409188764695813220925047059301953}{1504561126445636703796472818405047312320027667}$

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

Class group and class number

$C_{39}$, which has order $39$ (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:  \( 9087089.31038 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T141:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n141
Character table for t20n141 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.5.160801.1, 10.10.847280917741568.1, 10.0.1732416427267.1, 10.0.56767821488685056.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$67$67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed