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Properties

Label 20.0.16097667994...3761.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 47^{8}\cdot 107^{2}$
Root discriminant $12.89$
Ramified primes $3, 47, 107$
Class number $1$
Class group Trivial
Galois group 20T141

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 2, -6, 6, 0, 1, 3, 2, -4, 2, -2, -2, -1, -1, 2, -3, 2, 1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + x^18 + 2*x^17 - 3*x^16 + 2*x^15 - x^14 - x^13 - 2*x^12 - 2*x^11 + 2*x^10 - 4*x^9 + 2*x^8 + 3*x^7 + x^6 + 6*x^4 - 6*x^3 + 2*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^20 - x^19 + x^18 + 2*x^17 - 3*x^16 + 2*x^15 - x^14 - x^13 - 2*x^12 - 2*x^11 + 2*x^10 - 4*x^9 + 2*x^8 + 3*x^7 + x^6 + 6*x^4 - 6*x^3 + 2*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + x^{18} + 2 x^{17} - 3 x^{16} + 2 x^{15} - x^{14} - x^{13} - 2 x^{12} - 2 x^{11} + 2 x^{10} - 4 x^{9} + 2 x^{8} + 3 x^{7} + x^{6} + 6 x^{4} - 6 x^{3} + 2 x^{2} - 2 x + 1 \)

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:  \(16097667994068134233761=3^{10}\cdot 47^{8}\cdot 107^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.89$
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:  $3, 47, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{29} a^{18} - \frac{14}{29} a^{17} + \frac{14}{29} a^{16} + \frac{11}{29} a^{15} + \frac{11}{29} a^{14} + \frac{1}{29} a^{13} + \frac{12}{29} a^{12} - \frac{7}{29} a^{11} + \frac{4}{29} a^{10} - \frac{2}{29} a^{9} - \frac{10}{29} a^{8} + \frac{10}{29} a^{6} - \frac{11}{29} a^{5} - \frac{9}{29} a^{4} + \frac{4}{29} a^{3} - \frac{4}{29} a^{2} + \frac{8}{29} a - \frac{6}{29}$, $\frac{1}{201463} a^{19} - \frac{1255}{201463} a^{18} + \frac{45431}{201463} a^{17} + \frac{85239}{201463} a^{16} + \frac{45462}{201463} a^{15} - \frac{85628}{201463} a^{14} - \frac{92579}{201463} a^{13} - \frac{66722}{201463} a^{12} + \frac{55294}{201463} a^{11} - \frac{14565}{201463} a^{10} + \frac{63372}{201463} a^{9} - \frac{57335}{201463} a^{8} - \frac{51987}{201463} a^{7} - \frac{82311}{201463} a^{6} + \frac{96727}{201463} a^{5} + \frac{75379}{201463} a^{4} + \frac{16463}{201463} a^{3} + \frac{36611}{201463} a^{2} + \frac{92842}{201463} a + \frac{84267}{201463}$

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:  \( -\frac{5516}{6947} a^{19} + \frac{3368}{6947} a^{18} - \frac{5212}{6947} a^{17} - \frac{12311}{6947} a^{16} + \frac{11361}{6947} a^{15} - \frac{9429}{6947} a^{14} + \frac{5688}{6947} a^{13} + \frac{7333}{6947} a^{12} + \frac{13278}{6947} a^{11} + \frac{19326}{6947} a^{10} - \frac{806}{6947} a^{9} + \frac{25473}{6947} a^{8} - \frac{4921}{6947} a^{7} - \frac{14550}{6947} a^{6} - \frac{9585}{6947} a^{5} - \frac{12614}{6947} a^{4} - \frac{40406}{6947} a^{3} + \frac{16908}{6947} a^{2} - \frac{11420}{6947} a + \frac{13945}{6947} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1518.88671395 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T141:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n141
Character table for t20n141 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.1.2209.1, 10.0.1185762483.1, 10.2.126876585681.1, 10.0.522125867.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$47$47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$107$107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$