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Properties

Label 20.0.68622667523...6001.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 3409004107^{2}$
Root discriminant $15.55$
Ramified primes $3, 3409004107$
Class number $1$ (GRH)
Class group Trivial (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, 3, 9, -2, 0, 15, 12, -1, 5, 10, 14, -2, 1, 4, 6, 2, 2, -2, 1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + x^18 - 2*x^17 + 2*x^16 + 2*x^15 + 6*x^14 + 4*x^13 + x^12 - 2*x^11 + 14*x^10 + 10*x^9 + 5*x^8 - x^7 + 12*x^6 + 15*x^5 - 2*x^3 + 9*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^20 + x^18 - 2*x^17 + 2*x^16 + 2*x^15 + 6*x^14 + 4*x^13 + x^12 - 2*x^11 + 14*x^10 + 10*x^9 + 5*x^8 - x^7 + 12*x^6 + 15*x^5 - 2*x^3 + 9*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} + x^{18} - 2 x^{17} + 2 x^{16} + 2 x^{15} + 6 x^{14} + 4 x^{13} + x^{12} - 2 x^{11} + 14 x^{10} + 10 x^{9} + 5 x^{8} - x^{7} + 12 x^{6} + 15 x^{5} - 2 x^{3} + 9 x^{2} + 3 x + 1 \)

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:  \(686226675232104779996001=3^{10}\cdot 3409004107^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.55$
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:  $3, 3409004107$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{43} a^{18} - \frac{8}{43} a^{17} + \frac{1}{43} a^{16} - \frac{14}{43} a^{15} + \frac{7}{43} a^{14} - \frac{18}{43} a^{13} + \frac{3}{43} a^{12} + \frac{14}{43} a^{11} - \frac{2}{43} a^{10} + \frac{21}{43} a^{9} + \frac{17}{43} a^{8} - \frac{8}{43} a^{7} + \frac{13}{43} a^{6} + \frac{20}{43} a^{5} + \frac{9}{43} a^{4} - \frac{4}{43} a^{3} + \frac{15}{43} a^{2} + \frac{5}{43} a - \frac{2}{43}$, $\frac{1}{4314196235} a^{19} + \frac{37601584}{4314196235} a^{18} - \frac{18011411}{100330145} a^{17} + \frac{1411074726}{4314196235} a^{16} + \frac{692763341}{4314196235} a^{15} - \frac{421598059}{4314196235} a^{14} + \frac{65824097}{862839247} a^{13} + \frac{372781629}{4314196235} a^{12} - \frac{1507981588}{4314196235} a^{11} + \frac{1911979616}{4314196235} a^{10} - \frac{1107461062}{4314196235} a^{9} - \frac{311428}{4314196235} a^{8} - \frac{2059501027}{4314196235} a^{7} + \frac{688174736}{4314196235} a^{6} + \frac{2023963206}{4314196235} a^{5} + \frac{1697428034}{4314196235} a^{4} + \frac{401517481}{4314196235} a^{3} + \frac{528421777}{4314196235} a^{2} + \frac{1148411797}{4314196235} a - \frac{1187547179}{4314196235}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{24541734}{100330145} a^{19} - \frac{7112019}{100330145} a^{18} + \frac{23751278}{100330145} a^{17} - \frac{49742121}{100330145} a^{16} + \frac{63164259}{100330145} a^{15} + \frac{46501564}{100330145} a^{14} + \frac{23011704}{20066029} a^{13} + \frac{67460481}{100330145} a^{12} - \frac{2653662}{100330145} a^{11} - \frac{2596356}{100330145} a^{10} + \frac{395919927}{100330145} a^{9} + \frac{167309948}{100330145} a^{8} + \frac{15950637}{100330145} a^{7} + \frac{183654}{100330145} a^{6} + \frac{352258539}{100330145} a^{5} + \frac{420426066}{100330145} a^{4} - \frac{109467591}{100330145} a^{3} - \frac{4572677}{100330145} a^{2} + \frac{305907828}{100330145} a + \frac{104460429}{100330145} \) (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:  \( 9979.28108523 \) (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

\(\Q(\sqrt{-3}) \), 10.4.3409004107.1

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 ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
3Data not computed
3409004107Data not computed