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Properties

Label 20.0.14690182736...4129.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 61^{2}\cdot 401^{8}$
Root discriminant $28.73$
Ramified primes $3, 61, 401$
Class number $8$
Class group $[2, 4]$
Galois group 20T141

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 33, 36, 222, 51, 1283, -371, 3010, -740, 4764, -1801, 2611, -604, 778, -167, 147, -20, 16, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 16*x^18 - 20*x^17 + 147*x^16 - 167*x^15 + 778*x^14 - 604*x^13 + 2611*x^12 - 1801*x^11 + 4764*x^10 - 740*x^9 + 3010*x^8 - 371*x^7 + 1283*x^6 + 51*x^5 + 222*x^4 + 36*x^3 + 33*x^2 + 5*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 16*x^18 - 20*x^17 + 147*x^16 - 167*x^15 + 778*x^14 - 604*x^13 + 2611*x^12 - 1801*x^11 + 4764*x^10 - 740*x^9 + 3010*x^8 - 371*x^7 + 1283*x^6 + 51*x^5 + 222*x^4 + 36*x^3 + 33*x^2 + 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 16 x^{18} - 20 x^{17} + 147 x^{16} - 167 x^{15} + 778 x^{14} - 604 x^{13} + 2611 x^{12} - 1801 x^{11} + 4764 x^{10} - 740 x^{9} + 3010 x^{8} - 371 x^{7} + 1283 x^{6} + 51 x^{5} + 222 x^{4} + 36 x^{3} + 33 x^{2} + 5 x + 1 \)

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:  \(146901827368217141626017894129=3^{10}\cdot 61^{2}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.73$
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:  $3, 61, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{159} a^{18} + \frac{4}{53} a^{17} - \frac{4}{53} a^{16} + \frac{4}{159} a^{15} + \frac{11}{159} a^{14} + \frac{17}{53} a^{13} - \frac{28}{159} a^{12} - \frac{7}{53} a^{11} + \frac{14}{159} a^{10} + \frac{20}{159} a^{9} + \frac{74}{159} a^{8} - \frac{20}{159} a^{7} - \frac{61}{159} a^{6} + \frac{15}{53} a^{5} + \frac{12}{53} a^{4} + \frac{56}{159} a^{3} + \frac{15}{53} a^{2} + \frac{26}{53} a + \frac{43}{159}$, $\frac{1}{2261613136029127476416307} a^{19} - \frac{469757476826927729561}{251290348447680830712923} a^{18} + \frac{3923802143423283811073}{133036066825242792730371} a^{17} + \frac{42920800603751738143643}{753871045343042492138769} a^{16} + \frac{42882582835956388361777}{753871045343042492138769} a^{15} + \frac{3221260100634512223556}{119032270317322498758753} a^{14} - \frac{130747903992176092210303}{753871045343042492138769} a^{13} - \frac{788565486093732419220970}{2261613136029127476416307} a^{12} + \frac{1079252879748559588600532}{2261613136029127476416307} a^{11} + \frac{63985608016976330193140}{251290348447680830712923} a^{10} - \frac{60021880548134427966410}{251290348447680830712923} a^{9} + \frac{48017237750943712401868}{119032270317322498758753} a^{8} - \frac{312137997993443720136004}{753871045343042492138769} a^{7} + \frac{787929695792964527072416}{2261613136029127476416307} a^{6} - \frac{898106023383260035264256}{2261613136029127476416307} a^{5} + \frac{28510031681999381808977}{2261613136029127476416307} a^{4} - \frac{8339658416201852843072}{119032270317322498758753} a^{3} - \frac{282411992930287777163596}{2261613136029127476416307} a^{2} + \frac{1038419648087810653636987}{2261613136029127476416307} a - \frac{133636376633388072976868}{2261613136029127476416307}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$

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:  \( \frac{2725182770089811032379}{14223981987604575323373} a^{19} - \frac{5359375879495083421108}{14223981987604575323373} a^{18} + \frac{849241094313809514625}{278901607600089712223} a^{17} - \frac{17635754986199790480822}{4741327329201525107791} a^{16} + \frac{397203950532692712176323}{14223981987604575323373} a^{15} - \frac{23201079009111336572987}{748630630926556595967} a^{14} + \frac{2090719175527768498584130}{14223981987604575323373} a^{13} - \frac{1568629594869333460295254}{14223981987604575323373} a^{12} + \frac{6990730849904475772133333}{14223981987604575323373} a^{11} - \frac{4666579466869253398505807}{14223981987604575323373} a^{10} + \frac{4197222927341278547481318}{4741327329201525107791} a^{9} - \frac{27924572448110361427071}{249543543642185531989} a^{8} + \frac{7792078294447216006141264}{14223981987604575323373} a^{7} - \frac{1030248463857661452767797}{14223981987604575323373} a^{6} + \frac{3244590393754866129856982}{14223981987604575323373} a^{5} + \frac{35166071678595353113843}{4741327329201525107791} a^{4} + \frac{28054341113029144588612}{748630630926556595967} a^{3} + \frac{5010142822392403535721}{4741327329201525107791} a^{2} + \frac{25675840860757922979613}{4741327329201525107791} a + \frac{11544091286285126228639}{14223981987604575323373} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 842405.995326 \)
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

\(\Q(\sqrt{-3}) \), 5.5.160801.1, 10.0.6283241669043.1, 10.0.42586415756847.1, 10.10.14195471918949.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$61$61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
401Data not computed