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Properties

Label 20.0.28450861599...8064.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 11^{16}$
Root discriminant $23.59$
Ramified primes $2, 3, 11$
Class number $1$
Class group Trivial
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![1, 0, -15, 0, 190, 0, -469, 0, 796, 0, -711, 0, 454, 0, -182, 0, 53, 0, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^18 + 53*x^16 - 182*x^14 + 454*x^12 - 711*x^10 + 796*x^8 - 469*x^6 + 190*x^4 - 15*x^2 + 1)
 
gp: K = bnfinit(x^20 - 9*x^18 + 53*x^16 - 182*x^14 + 454*x^12 - 711*x^10 + 796*x^8 - 469*x^6 + 190*x^4 - 15*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{18} + 53 x^{16} - 182 x^{14} + 454 x^{12} - 711 x^{10} + 796 x^{8} - 469 x^{6} + 190 x^{4} - 15 x^{2} + 1 \)

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:  \(2845086159957207322343768064=2^{20}\cdot 3^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.59$
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$$\chi_{132}(1,·)$, $\chi_{132}(67,·)$, $\chi_{132}(5,·)$, $\chi_{132}(71,·)$, $\chi_{132}(23,·)$, $\chi_{132}(25,·)$, $\chi_{132}(91,·)$, $\chi_{132}(31,·)$, $\chi_{132}(97,·)$, $\chi_{132}(113,·)$, $\chi_{132}(37,·)$, $\chi_{132}(103,·)$, $\chi_{132}(47,·)$, $\chi_{132}(49,·)$, $\chi_{132}(115,·)$, $\chi_{132}(53,·)$, $\chi_{132}(89,·)$, $\chi_{132}(119,·)$, $\chi_{132}(59,·)$, $\chi_{132}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{1752049531} a^{18} + \frac{335285807}{1752049531} a^{16} + \frac{44712727}{1752049531} a^{14} - \frac{57936275}{1752049531} a^{12} + \frac{189558270}{1752049531} a^{10} - \frac{205566055}{1752049531} a^{8} - \frac{70533826}{1752049531} a^{6} - \frac{289478054}{1752049531} a^{4} - \frac{652067004}{1752049531} a^{2} - \frac{96297369}{1752049531}$, $\frac{1}{1752049531} a^{19} + \frac{335285807}{1752049531} a^{17} + \frac{44712727}{1752049531} a^{15} - \frac{57936275}{1752049531} a^{13} + \frac{189558270}{1752049531} a^{11} - \frac{205566055}{1752049531} a^{9} - \frac{70533826}{1752049531} a^{7} - \frac{289478054}{1752049531} a^{5} - \frac{652067004}{1752049531} a^{3} - \frac{96297369}{1752049531} a$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{159474569}{1752049531} a^{19} - \frac{1374428400}{1752049531} a^{17} + \frac{7884637588}{1752049531} a^{15} - \frac{25646086180}{1752049531} a^{13} + \frac{60534408164}{1752049531} a^{11} - \frac{83643130996}{1752049531} a^{9} + \frac{79807475842}{1752049531} a^{7} - \frac{23759285608}{1752049531} a^{5} + \frac{1882966908}{1752049531} a^{3} + \frac{6041832875}{1752049531} a \) (order $12$)
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:  \( 873105.021385 \)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{12})\), \(\Q(\zeta_{11})^+\), 10.0.219503494144.1, 10.10.53339349076992.1, 10.0.52089208083.1

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 R R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$