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Properties

Label 20.20.2845893327...1117.1
Degree $20$
Signature $[20, 0]$
Discriminant $11^{18}\cdot 13^{15}$
Root discriminant $59.25$
Ramified primes $11, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-43, -1316, -10254, 1845, 78872, 16639, -194675, -51088, 224814, 54957, -139547, -27812, 48842, 7010, -9572, -854, 998, 48, -51, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 51*x^18 + 48*x^17 + 998*x^16 - 854*x^15 - 9572*x^14 + 7010*x^13 + 48842*x^12 - 27812*x^11 - 139547*x^10 + 54957*x^9 + 224814*x^8 - 51088*x^7 - 194675*x^6 + 16639*x^5 + 78872*x^4 + 1845*x^3 - 10254*x^2 - 1316*x - 43)
 
gp: K = bnfinit(x^20 - x^19 - 51*x^18 + 48*x^17 + 998*x^16 - 854*x^15 - 9572*x^14 + 7010*x^13 + 48842*x^12 - 27812*x^11 - 139547*x^10 + 54957*x^9 + 224814*x^8 - 51088*x^7 - 194675*x^6 + 16639*x^5 + 78872*x^4 + 1845*x^3 - 10254*x^2 - 1316*x - 43, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 51 x^{18} + 48 x^{17} + 998 x^{16} - 854 x^{15} - 9572 x^{14} + 7010 x^{13} + 48842 x^{12} - 27812 x^{11} - 139547 x^{10} + 54957 x^{9} + 224814 x^{8} - 51088 x^{7} - 194675 x^{6} + 16639 x^{5} + 78872 x^{4} + 1845 x^{3} - 10254 x^{2} - 1316 x - 43 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(284589332775604260722209388186521117=11^{18}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.25$
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:  $11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(143=11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{143}(64,·)$, $\chi_{143}(1,·)$, $\chi_{143}(8,·)$, $\chi_{143}(73,·)$, $\chi_{143}(138,·)$, $\chi_{143}(12,·)$, $\chi_{143}(14,·)$, $\chi_{143}(18,·)$, $\chi_{143}(83,·)$, $\chi_{143}(21,·)$, $\chi_{143}(25,·)$, $\chi_{143}(27,·)$, $\chi_{143}(92,·)$, $\chi_{143}(96,·)$, $\chi_{143}(38,·)$, $\chi_{143}(103,·)$, $\chi_{143}(109,·)$, $\chi_{143}(112,·)$, $\chi_{143}(53,·)$, $\chi_{143}(57,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{419} a^{15} - \frac{204}{419} a^{14} + \frac{180}{419} a^{13} - \frac{150}{419} a^{12} + \frac{146}{419} a^{11} - \frac{141}{419} a^{10} + \frac{108}{419} a^{9} - \frac{162}{419} a^{8} + \frac{204}{419} a^{7} - \frac{75}{419} a^{6} - \frac{143}{419} a^{5} - \frac{86}{419} a^{4} + \frac{76}{419} a^{3} + \frac{49}{419} a^{2} + \frac{22}{419} a - \frac{91}{419}$, $\frac{1}{419} a^{16} + \frac{45}{419} a^{14} + \frac{117}{419} a^{13} + \frac{133}{419} a^{12} - \frac{106}{419} a^{11} - \frac{164}{419} a^{10} + \frac{82}{419} a^{9} - \frac{162}{419} a^{8} + \frac{60}{419} a^{7} + \frac{60}{419} a^{6} + \frac{72}{419} a^{5} + \frac{130}{419} a^{4} + \frac{50}{419} a^{3} - \frac{38}{419} a^{2} + \frac{207}{419} a - \frac{128}{419}$, $\frac{1}{419} a^{17} + \frac{79}{419} a^{14} - \frac{6}{419} a^{13} - \frac{60}{419} a^{12} - \frac{30}{419} a^{11} + \frac{142}{419} a^{10} + \frac{6}{419} a^{9} - \frac{192}{419} a^{8} + \frac{98}{419} a^{7} + \frac{95}{419} a^{6} - \frac{139}{419} a^{5} + \frac{149}{419} a^{4} - \frac{106}{419} a^{3} + \frac{97}{419} a^{2} + \frac{139}{419} a - \frac{95}{419}$, $\frac{1}{414391} a^{18} + \frac{24}{414391} a^{17} - \frac{381}{414391} a^{16} - \frac{149}{414391} a^{15} - \frac{34107}{414391} a^{14} + \frac{120746}{414391} a^{13} - \frac{188057}{414391} a^{12} + \frac{138086}{414391} a^{11} - \frac{382}{989} a^{10} + \frac{77747}{414391} a^{9} - \frac{2858}{9637} a^{8} + \frac{71345}{414391} a^{7} + \frac{29063}{414391} a^{6} - \frac{182794}{414391} a^{5} + \frac{179277}{414391} a^{4} + \frac{29472}{414391} a^{3} - \frac{180263}{414391} a^{2} - \frac{156900}{414391} a - \frac{3250}{9637}$, $\frac{1}{551707846063147927851149} a^{19} + \frac{206805579424702201}{551707846063147927851149} a^{18} - \frac{137915729889006927592}{551707846063147927851149} a^{17} - \frac{62902355965607763829}{551707846063147927851149} a^{16} - \frac{249888182928191035170}{551707846063147927851149} a^{15} + \frac{241947575607220123805806}{551707846063147927851149} a^{14} - \frac{264229343149481093409270}{551707846063147927851149} a^{13} - \frac{11468738457150148998352}{23987297654919475123963} a^{12} + \frac{218288813186859073602766}{551707846063147927851149} a^{11} - \frac{5321952713506791425168}{23987297654919475123963} a^{10} - \frac{210088494242919206077227}{551707846063147927851149} a^{9} - \frac{40665350294268760269183}{551707846063147927851149} a^{8} + \frac{63054173454947701141391}{551707846063147927851149} a^{7} - \frac{236394366831530464340311}{551707846063147927851149} a^{6} + \frac{209452542004172816672513}{551707846063147927851149} a^{5} + \frac{243971176879060301130255}{551707846063147927851149} a^{4} + \frac{275235933190421439913163}{551707846063147927851149} a^{3} - \frac{129777801486976327854157}{551707846063147927851149} a^{2} + \frac{149085428811512675900137}{551707846063147927851149} a + \frac{5476076252065010346817}{12830415024724370415143}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $19$
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:  \( 168224843913 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

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

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.265837.1, \(\Q(\zeta_{11})^+\), 10.10.79589952003133.1

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 $20$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ $20$ $20$ R R ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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
11Data not computed
13Data not computed