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Properties

Label 20.14.5906964566...4224.1
Degree $20$
Signature $[14, 3]$
Discriminant $-\,2^{42}\cdot 7^{8}\cdot 13^{12}$
Root discriminant $43.51$
Ramified primes $2, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T633

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 66, 0, -579, 0, 1582, 0, -942, 0, -410, 0, 429, 0, -12, 0, -44, 0, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 4*x^18 - 44*x^16 - 12*x^14 + 429*x^12 - 410*x^10 - 942*x^8 + 1582*x^6 - 579*x^4 + 66*x^2 - 1)
 
gp: K = bnfinit(x^20 + 4*x^18 - 44*x^16 - 12*x^14 + 429*x^12 - 410*x^10 - 942*x^8 + 1582*x^6 - 579*x^4 + 66*x^2 - 1, 1)
 

Normalized defining polynomial

\( x^{20} + 4 x^{18} - 44 x^{16} - 12 x^{14} + 429 x^{12} - 410 x^{10} - 942 x^{8} + 1582 x^{6} - 579 x^{4} + 66 x^{2} - 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-590696456616395826333497084084224=-\,2^{42}\cdot 7^{8}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.51$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{26} a^{12} - \frac{1}{2} a^{10} - \frac{11}{26} a^{8} - \frac{11}{26} a^{6} + \frac{5}{26} a^{4} - \frac{4}{13} a^{2} - \frac{1}{26}$, $\frac{1}{26} a^{13} - \frac{1}{2} a^{11} - \frac{11}{26} a^{9} - \frac{11}{26} a^{7} + \frac{5}{26} a^{5} - \frac{4}{13} a^{3} - \frac{1}{26} a$, $\frac{1}{26} a^{14} + \frac{1}{13} a^{10} + \frac{1}{13} a^{8} - \frac{4}{13} a^{6} + \frac{5}{26} a^{4} - \frac{1}{26} a^{2} - \frac{1}{2}$, $\frac{1}{26} a^{15} + \frac{1}{13} a^{11} + \frac{1}{13} a^{9} - \frac{4}{13} a^{7} + \frac{5}{26} a^{5} - \frac{1}{26} a^{3} - \frac{1}{2} a$, $\frac{1}{52} a^{16} - \frac{1}{52} a^{15} - \frac{1}{52} a^{14} - \frac{1}{52} a^{13} + \frac{11}{52} a^{11} - \frac{1}{2} a^{10} + \frac{9}{52} a^{9} + \frac{3}{13} a^{8} + \frac{19}{52} a^{7} + \frac{9}{52} a^{6} + \frac{4}{13} a^{5} - \frac{4}{13} a^{4} + \frac{9}{52} a^{3} - \frac{11}{26} a^{2} - \frac{3}{13} a + \frac{15}{52}$, $\frac{1}{52} a^{17} - \frac{1}{52} a^{13} - \frac{1}{52} a^{12} - \frac{11}{52} a^{11} - \frac{1}{4} a^{10} + \frac{25}{52} a^{9} + \frac{11}{52} a^{8} + \frac{3}{13} a^{7} - \frac{15}{52} a^{6} + \frac{5}{26} a^{5} - \frac{5}{52} a^{4} - \frac{15}{52} a^{3} + \frac{2}{13} a^{2} - \frac{23}{52} a + \frac{1}{52}$, $\frac{1}{51532} a^{18} - \frac{111}{12883} a^{16} - \frac{323}{51532} a^{14} - \frac{1}{52} a^{13} - \frac{985}{51532} a^{12} - \frac{1}{4} a^{11} - \frac{20097}{51532} a^{10} + \frac{11}{52} a^{9} + \frac{8329}{25766} a^{8} - \frac{15}{52} a^{7} - \frac{374}{12883} a^{6} - \frac{5}{52} a^{5} + \frac{18721}{51532} a^{4} + \frac{2}{13} a^{3} - \frac{9673}{51532} a^{2} + \frac{1}{52} a - \frac{6212}{12883}$, $\frac{1}{51532} a^{19} - \frac{111}{12883} a^{17} - \frac{323}{51532} a^{15} - \frac{1}{52} a^{14} - \frac{985}{51532} a^{13} - \frac{1}{52} a^{12} - \frac{20097}{51532} a^{11} + \frac{11}{52} a^{10} + \frac{8329}{25766} a^{9} + \frac{9}{52} a^{8} - \frac{374}{12883} a^{7} + \frac{19}{52} a^{6} + \frac{18721}{51532} a^{5} + \frac{4}{13} a^{4} - \frac{9673}{51532} a^{3} + \frac{9}{52} a^{2} - \frac{6212}{12883} a - \frac{3}{13}$

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:  $16$
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:  \( 5283743002.08 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T633:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40960
The 124 conjugacy class representatives for t20n633 are not computed
Character table for t20n633 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.5.6889792.1, 10.10.379753870426112.1

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 R $20$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$