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Properties

Label 20.20.2003171323...6608.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{55}\cdot 11^{18}$
Root discriminant $58.22$
Ramified primes $2, 11$
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![3872, 0, -38720, 0, 129712, 0, -212960, 0, 196504, 0, -107712, 0, 35552, 0, -6952, 0, 770, 0, -44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 44*x^18 + 770*x^16 - 6952*x^14 + 35552*x^12 - 107712*x^10 + 196504*x^8 - 212960*x^6 + 129712*x^4 - 38720*x^2 + 3872)
 
gp: K = bnfinit(x^20 - 44*x^18 + 770*x^16 - 6952*x^14 + 35552*x^12 - 107712*x^10 + 196504*x^8 - 212960*x^6 + 129712*x^4 - 38720*x^2 + 3872, 1)
 

Normalized defining polynomial

\( x^{20} - 44 x^{18} + 770 x^{16} - 6952 x^{14} + 35552 x^{12} - 107712 x^{10} + 196504 x^{8} - 212960 x^{6} + 129712 x^{4} - 38720 x^{2} + 3872 \)

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:  \(200317132330035063121671003054276608=2^{55}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.22$
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:  $2, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(176=2^{4}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{176}(1,·)$, $\chi_{176}(131,·)$, $\chi_{176}(81,·)$, $\chi_{176}(9,·)$, $\chi_{176}(139,·)$, $\chi_{176}(43,·)$, $\chi_{176}(83,·)$, $\chi_{176}(171,·)$, $\chi_{176}(25,·)$, $\chi_{176}(89,·)$, $\chi_{176}(51,·)$, $\chi_{176}(97,·)$, $\chi_{176}(35,·)$, $\chi_{176}(113,·)$, $\chi_{176}(169,·)$, $\chi_{176}(107,·)$, $\chi_{176}(49,·)$, $\chi_{176}(19,·)$, $\chi_{176}(137,·)$, $\chi_{176}(123,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{44} a^{10}$, $\frac{1}{44} a^{11}$, $\frac{1}{88} a^{12}$, $\frac{1}{88} a^{13}$, $\frac{1}{88} a^{14}$, $\frac{1}{88} a^{15}$, $\frac{1}{4048} a^{16} - \frac{1}{2024} a^{14} + \frac{5}{2024} a^{12} - \frac{7}{1012} a^{10} - \frac{2}{23} a^{8} - \frac{3}{23} a^{6} - \frac{7}{46} a^{4} + \frac{4}{23} a^{2} + \frac{5}{23}$, $\frac{1}{4048} a^{17} - \frac{1}{2024} a^{15} + \frac{5}{2024} a^{13} - \frac{7}{1012} a^{11} - \frac{2}{23} a^{9} - \frac{3}{23} a^{7} - \frac{7}{46} a^{5} + \frac{4}{23} a^{3} + \frac{5}{23} a$, $\frac{1}{805552} a^{18} + \frac{45}{805552} a^{16} + \frac{945}{201388} a^{14} + \frac{87}{36616} a^{12} - \frac{340}{50347} a^{10} - \frac{396}{4577} a^{8} - \frac{197}{9154} a^{6} - \frac{126}{4577} a^{4} - \frac{129}{4577} a^{2} - \frac{1950}{4577}$, $\frac{1}{805552} a^{19} + \frac{45}{805552} a^{17} + \frac{945}{201388} a^{15} + \frac{87}{36616} a^{13} - \frac{340}{50347} a^{11} - \frac{396}{4577} a^{9} - \frac{197}{9154} a^{7} - \frac{126}{4577} a^{5} - \frac{129}{4577} a^{3} - \frac{1950}{4577} a$

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:  \( 228779211109 \) (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

\(\Q(\sqrt{2}) \), 4.4.247808.1, \(\Q(\zeta_{11})^+\), 10.10.7024111812608.1

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 $20$ $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ $20$ $20$

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
2Data not computed
11Data not computed