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Properties

Label 20.0.11543661841...7824.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 401^{10}$
Root discriminant $56.64$
Ramified primes $2, 401$
Class number $5046$ (GRH)
Class group $[29, 174]$ (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1497569, 204212, 2333308, 462756, 1783158, 418974, 856214, 230252, 287947, 89262, 70881, 23190, 13433, 4112, 1750, 378, 222, 12, -16, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 16*x^18 + 12*x^17 + 222*x^16 + 378*x^15 + 1750*x^14 + 4112*x^13 + 13433*x^12 + 23190*x^11 + 70881*x^10 + 89262*x^9 + 287947*x^8 + 230252*x^7 + 856214*x^6 + 418974*x^5 + 1783158*x^4 + 462756*x^3 + 2333308*x^2 + 204212*x + 1497569)
 
gp: K = bnfinit(x^20 - 4*x^19 - 16*x^18 + 12*x^17 + 222*x^16 + 378*x^15 + 1750*x^14 + 4112*x^13 + 13433*x^12 + 23190*x^11 + 70881*x^10 + 89262*x^9 + 287947*x^8 + 230252*x^7 + 856214*x^6 + 418974*x^5 + 1783158*x^4 + 462756*x^3 + 2333308*x^2 + 204212*x + 1497569, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 16 x^{18} + 12 x^{17} + 222 x^{16} + 378 x^{15} + 1750 x^{14} + 4112 x^{13} + 13433 x^{12} + 23190 x^{11} + 70881 x^{10} + 89262 x^{9} + 287947 x^{8} + 230252 x^{7} + 856214 x^{6} + 418974 x^{5} + 1783158 x^{4} + 462756 x^{3} + 2333308 x^{2} + 204212 x + 1497569 \)

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:  \(115436618418847759115953789493837824=2^{30}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.64$
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:  $2, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{10} + \frac{1}{9} a^{6} + \frac{2}{9} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} + \frac{1}{9} a^{7} + \frac{2}{9} a^{3} + \frac{1}{9} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{10} + \frac{1}{9} a^{8} + \frac{1}{9} a^{6} + \frac{2}{9} a^{4} + \frac{1}{3} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{11} + \frac{1}{9} a^{9} + \frac{1}{9} a^{7} + \frac{2}{9} a^{5} + \frac{1}{3} a^{3} + \frac{1}{9} a$, $\frac{1}{81} a^{16} - \frac{1}{27} a^{15} - \frac{1}{81} a^{14} + \frac{1}{27} a^{13} + \frac{1}{81} a^{12} - \frac{4}{27} a^{11} + \frac{1}{27} a^{9} + \frac{1}{9} a^{8} + \frac{1}{27} a^{7} + \frac{4}{27} a^{6} - \frac{1}{27} a^{5} - \frac{17}{81} a^{4} + \frac{4}{27} a^{3} + \frac{11}{81} a^{2} - \frac{1}{27} a - \frac{26}{81}$, $\frac{1}{81} a^{17} - \frac{1}{81} a^{15} + \frac{1}{81} a^{13} - \frac{1}{9} a^{11} - \frac{2}{27} a^{10} + \frac{1}{27} a^{8} - \frac{2}{27} a^{7} - \frac{4}{27} a^{6} - \frac{35}{81} a^{5} - \frac{4}{27} a^{4} + \frac{2}{81} a^{3} - \frac{2}{27} a^{2} - \frac{35}{81} a - \frac{5}{27}$, $\frac{1}{3159} a^{18} - \frac{5}{1053} a^{17} + \frac{4}{1053} a^{16} - \frac{44}{1053} a^{15} - \frac{55}{1053} a^{14} + \frac{38}{1053} a^{13} + \frac{10}{243} a^{12} - \frac{55}{351} a^{11} - \frac{20}{351} a^{10} - \frac{166}{1053} a^{9} - \frac{101}{1053} a^{8} - \frac{58}{351} a^{7} - \frac{239}{3159} a^{6} + \frac{26}{81} a^{5} + \frac{335}{1053} a^{4} + \frac{487}{1053} a^{3} + \frac{11}{351} a^{2} - \frac{47}{1053} a + \frac{211}{3159}$, $\frac{1}{4622622013433723557839120206012538748450020087} a^{19} - \frac{204012882564854622185756306162134692266161}{4622622013433723557839120206012538748450020087} a^{18} - \frac{213660789585975930567132040843113748911115}{171208222719767539179226674296760694387037781} a^{17} - \frac{8825070444788564411125233311798407047170117}{1540874004477907852613040068670846249483340029} a^{16} - \frac{11165235893898027442828027168165657922410460}{1540874004477907852613040068670846249483340029} a^{15} + \frac{10958875094033743214871708736153850740750778}{1540874004477907852613040068670846249483340029} a^{14} - \frac{108056567873376776225293150666085140780238681}{4622622013433723557839120206012538748450020087} a^{13} + \frac{121846648292001061997346325430297829234277886}{4622622013433723557839120206012538748450020087} a^{12} + \frac{25957647404461607123186498463622512885258943}{513624668159302617537680022890282083161113343} a^{11} + \frac{152693641800869951136831769417235478491821124}{1540874004477907852613040068670846249483340029} a^{10} + \frac{72705056866566787514547367456752942312323828}{1540874004477907852613040068670846249483340029} a^{9} + \frac{133240573384771003685244442809786370099495502}{1540874004477907852613040068670846249483340029} a^{8} + \frac{583522128941149922296343944703454316240262806}{4622622013433723557839120206012538748450020087} a^{7} + \frac{244781639913691986647919445878257032834949015}{4622622013433723557839120206012538748450020087} a^{6} - \frac{162819263179169069002296902109074060445881956}{513624668159302617537680022890282083161113343} a^{5} - \frac{13695838506354039230042085448029149149210981}{513624668159302617537680022890282083161113343} a^{4} - \frac{6549584563642203869442262857888744697246811}{18564747041902504248349880345431882523895663} a^{3} - \frac{244811961930464369474322497590411064864794298}{513624668159302617537680022890282083161113343} a^{2} + \frac{45254341861693549212233647805462089610787280}{124935730092803339401057302865203749958108651} a - \frac{8704400197926551918725854258820379805327452}{55694241125707512745049641036295647571686989}$

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

Class group and class number

$C_{29}\times C_{174}$, which has order $5046$ (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:  \( 795087.603907 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{401}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-802}) \), \(\Q(\sqrt{-2}, \sqrt{401})\), 5.5.160801.1 x5, 10.10.10368641602001.1, 10.0.847280917741568.1 x5, 10.0.339759648014368768.1 x5

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

Sibling fields

Degree 10 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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.15.9$x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.9$x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$$2$$5$$15$$C_{10}$$[3]^{5}$
401Data not computed