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Properties

Label 20.0.34425542535...5744.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 11^{18}$
Root discriminant $29.98$
Ramified primes $2, 3, 11$
Class number $22$ (GRH)
Class group $[22]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121, 0, 605, 0, 2178, 0, 3267, 0, 3388, 0, 2189, 0, 1034, 0, 330, 0, 77, 0, 11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 11*x^18 + 77*x^16 + 330*x^14 + 1034*x^12 + 2189*x^10 + 3388*x^8 + 3267*x^6 + 2178*x^4 + 605*x^2 + 121)
 
gp: K = bnfinit(x^20 + 11*x^18 + 77*x^16 + 330*x^14 + 1034*x^12 + 2189*x^10 + 3388*x^8 + 3267*x^6 + 2178*x^4 + 605*x^2 + 121, 1)
 

Normalized defining polynomial

\( x^{20} + 11 x^{18} + 77 x^{16} + 330 x^{14} + 1034 x^{12} + 2189 x^{10} + 3388 x^{8} + 3267 x^{6} + 2178 x^{4} + 605 x^{2} + 121 \)

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:  \(344255425354822086003595935744=2^{20}\cdot 3^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.98$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(132=2^{2}\cdot 3\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{132}(1,·)$, $\chi_{132}(131,·)$, $\chi_{132}(5,·)$, $\chi_{132}(7,·)$, $\chi_{132}(79,·)$, $\chi_{132}(43,·)$, $\chi_{132}(83,·)$, $\chi_{132}(25,·)$, $\chi_{132}(89,·)$, $\chi_{132}(95,·)$, $\chi_{132}(97,·)$, $\chi_{132}(35,·)$, $\chi_{132}(37,·)$, $\chi_{132}(113,·)$, $\chi_{132}(107,·)$, $\chi_{132}(49,·)$, $\chi_{132}(19,·)$, $\chi_{132}(53,·)$, $\chi_{132}(125,·)$, $\chi_{132}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10}$, $\frac{1}{11} a^{11}$, $\frac{1}{11} a^{12}$, $\frac{1}{11} a^{13}$, $\frac{1}{11} a^{14}$, $\frac{1}{11} a^{15}$, $\frac{1}{473} a^{16} - \frac{7}{473} a^{14} + \frac{6}{473} a^{12} + \frac{10}{473} a^{10} + \frac{21}{43} a^{8} - \frac{18}{43} a^{6} + \frac{21}{43} a^{4} - \frac{18}{43} a^{2} - \frac{3}{43}$, $\frac{1}{473} a^{17} - \frac{7}{473} a^{15} + \frac{6}{473} a^{13} + \frac{10}{473} a^{11} + \frac{21}{43} a^{9} - \frac{18}{43} a^{7} + \frac{21}{43} a^{5} - \frac{18}{43} a^{3} - \frac{3}{43} a$, $\frac{1}{3454319} a^{18} + \frac{1064}{3454319} a^{16} - \frac{62617}{3454319} a^{14} + \frac{11597}{314029} a^{12} + \frac{48824}{3454319} a^{10} - \frac{55701}{314029} a^{8} + \frac{34063}{314029} a^{6} + \frac{10906}{314029} a^{4} + \frac{142657}{314029} a^{2} + \frac{53590}{314029}$, $\frac{1}{3454319} a^{19} + \frac{1064}{3454319} a^{17} - \frac{62617}{3454319} a^{15} + \frac{11597}{314029} a^{13} + \frac{48824}{3454319} a^{11} - \frac{55701}{314029} a^{9} + \frac{34063}{314029} a^{7} + \frac{10906}{314029} a^{5} + \frac{142657}{314029} a^{3} + \frac{53590}{314029} a$

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

Class group and class number

$C_{22}$, which has order $22$ (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:  \( \frac{10843}{3454319} a^{18} + \frac{10460}{314029} a^{16} + \frac{792503}{3454319} a^{14} + \frac{3292525}{3454319} a^{12} + \frac{10082302}{3454319} a^{10} + \frac{1854422}{314029} a^{8} + \frac{2778327}{314029} a^{6} + \frac{2435481}{314029} a^{4} + \frac{1712232}{314029} a^{2} + \frac{473264}{314029} \) (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:  \( 281202.490766 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-33}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-3}, \sqrt{11})\), \(\Q(\zeta_{11})^+\), 10.0.586732839846912.1, 10.0.52089208083.1, \(\Q(\zeta_{44})^+\)

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 R R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
3Data not computed
11Data not computed