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Properties

Label 20.0.15545170849...1729.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 460181^{2}\cdot 1114969^{2}$
Root discriminant $25.68$
Ramified primes $3, 460181, 1114969$
Class number $11$ (GRH)
Class group $[11]$ (GRH)
Galois group 20T1021

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 18, -42, 294, -418, 959, -787, 1544, -1144, 1518, -910, 951, -512, 407, -174, 107, -38, 18, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 18*x^18 - 38*x^17 + 107*x^16 - 174*x^15 + 407*x^14 - 512*x^13 + 951*x^12 - 910*x^11 + 1518*x^10 - 1144*x^9 + 1544*x^8 - 787*x^7 + 959*x^6 - 418*x^5 + 294*x^4 - 42*x^3 + 18*x^2 + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 18*x^18 - 38*x^17 + 107*x^16 - 174*x^15 + 407*x^14 - 512*x^13 + 951*x^12 - 910*x^11 + 1518*x^10 - 1144*x^9 + 1544*x^8 - 787*x^7 + 959*x^6 - 418*x^5 + 294*x^4 - 42*x^3 + 18*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 18 x^{18} - 38 x^{17} + 107 x^{16} - 174 x^{15} + 407 x^{14} - 512 x^{13} + 951 x^{12} - 910 x^{11} + 1518 x^{10} - 1144 x^{9} + 1544 x^{8} - 787 x^{7} + 959 x^{6} - 418 x^{5} + 294 x^{4} - 42 x^{3} + 18 x^{2} + 1 \)

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:  \(15545170849776123808325331729=3^{10}\cdot 460181^{2}\cdot 1114969^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.68$
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:  $3, 460181, 1114969$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{159856743223057525457} a^{19} - \frac{10544961767872149724}{159856743223057525457} a^{18} - \frac{8551779304247761715}{159856743223057525457} a^{17} + \frac{10271146555773921497}{159856743223057525457} a^{16} - \frac{34197023906663594251}{159856743223057525457} a^{15} - \frac{21802597676458653314}{159856743223057525457} a^{14} + \frac{48155570273882698847}{159856743223057525457} a^{13} + \frac{76796305305961920469}{159856743223057525457} a^{12} + \frac{56666955056970901446}{159856743223057525457} a^{11} + \frac{2729039948471160672}{159856743223057525457} a^{10} - \frac{54636623002863106299}{159856743223057525457} a^{9} - \frac{69439515055967344780}{159856743223057525457} a^{8} - \frac{24776789784399660782}{159856743223057525457} a^{7} + \frac{14067628163326660288}{159856743223057525457} a^{6} + \frac{32396148174275834181}{159856743223057525457} a^{5} + \frac{70623775109551192673}{159856743223057525457} a^{4} + \frac{50740040386772382050}{159856743223057525457} a^{3} - \frac{64577324141370431313}{159856743223057525457} a^{2} - \frac{72084942586473911158}{159856743223057525457} a - \frac{29579092147840744526}{159856743223057525457}$

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

Class group and class number

$C_{11}$, which has order $11$ (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:  \( -\frac{178243254842046041}{159856743223057525457} a^{19} + \frac{11236567537781013694}{159856743223057525457} a^{18} - \frac{45765678357083597320}{159856743223057525457} a^{17} + \frac{197526769594523394275}{159856743223057525457} a^{16} - \frac{424677758861627084661}{159856743223057525457} a^{15} + \frac{1163244138199388216538}{159856743223057525457} a^{14} - \frac{1924674186773890939789}{159856743223057525457} a^{13} + \frac{4382878309543691562845}{159856743223057525457} a^{12} - \frac{5609668941497558349635}{159856743223057525457} a^{11} + \frac{10130249951756733579456}{159856743223057525457} a^{10} - \frac{9860484921391881421968}{159856743223057525457} a^{9} + \frac{15943029357171691158671}{159856743223057525457} a^{8} - \frac{12222930702440030851251}{159856743223057525457} a^{7} + \frac{15925500861601555262432}{159856743223057525457} a^{6} - \frac{8122261834268209502745}{159856743223057525457} a^{5} + \frac{9556822614393175263528}{159856743223057525457} a^{4} - \frac{4147267080611462928276}{159856743223057525457} a^{3} + \frac{2904455811233724923052}{159856743223057525457} a^{2} - \frac{209472997719917882982}{159856743223057525457} a + \frac{177766393511953500666}{159856743223057525457} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 114785.123876 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1021:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7257600
The 84 conjugacy class representatives for t20n1021 are not computed
Character table for t20n1021 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 10.10.513087549389.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 40 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $18{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $18{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $18{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
460181Data not computed
1114969Data not computed