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Properties

Label 20.10.2584296997...6848.4
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{38}\cdot 7^{9}\cdot 13^{12}$
Root discriminant $41.75$
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![-343, 0, -2730, 0, -2127, 0, 8346, 0, 3430, 0, -6994, 0, 1771, 0, 208, 0, -90, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 90*x^16 + 208*x^14 + 1771*x^12 - 6994*x^10 + 3430*x^8 + 8346*x^6 - 2127*x^4 - 2730*x^2 - 343)
 
gp: K = bnfinit(x^20 - 90*x^16 + 208*x^14 + 1771*x^12 - 6994*x^10 + 3430*x^8 + 8346*x^6 - 2127*x^4 - 2730*x^2 - 343, 1)
 

Normalized defining polynomial

\( x^{20} - 90 x^{16} + 208 x^{14} + 1771 x^{12} - 6994 x^{10} + 3430 x^{8} + 8346 x^{6} - 2127 x^{4} - 2730 x^{2} - 343 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-258429699769673174020904974286848=-\,2^{38}\cdot 7^{9}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.75$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{28} a^{14} - \frac{1}{4} a^{13} - \frac{3}{14} a^{12} - \frac{1}{4} a^{9} - \frac{1}{7} a^{8} - \frac{1}{2} a^{7} + \frac{5}{14} a^{6} - \frac{1}{4} a^{5} - \frac{11}{28} a^{4} + \frac{1}{4} a^{3} + \frac{11}{28} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{28} a^{15} + \frac{1}{28} a^{13} - \frac{1}{4} a^{10} + \frac{3}{28} a^{9} - \frac{1}{7} a^{7} + \frac{1}{4} a^{6} - \frac{1}{7} a^{5} + \frac{1}{7} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{364} a^{16} - \frac{1}{4} a^{13} + \frac{3}{182} a^{12} - \frac{1}{4} a^{11} - \frac{3}{28} a^{10} - \frac{1}{4} a^{9} - \frac{3}{13} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{97}{364} a^{4} - \frac{5}{14} a^{2} - \frac{1}{4} a + \frac{15}{52}$, $\frac{1}{364} a^{17} - \frac{85}{364} a^{13} - \frac{1}{4} a^{12} - \frac{3}{28} a^{11} - \frac{1}{4} a^{10} + \frac{1}{52} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{44}{91} a^{5} - \frac{1}{4} a^{4} - \frac{3}{28} a^{3} + \frac{15}{52} a - \frac{1}{4}$, $\frac{1}{51205892695048} a^{18} + \frac{27653993361}{51205892695048} a^{16} - \frac{778849093003}{51205892695048} a^{14} - \frac{1}{4} a^{13} + \frac{838270911507}{51205892695048} a^{12} + \frac{4585634870619}{25602946347524} a^{10} - \frac{1}{4} a^{9} - \frac{521456897678}{6400736586881} a^{8} - \frac{1}{2} a^{7} + \frac{8522457955983}{25602946347524} a^{6} - \frac{1}{4} a^{5} - \frac{8337333435049}{25602946347524} a^{4} + \frac{1}{4} a^{3} - \frac{6365276390537}{51205892695048} a^{2} - \frac{1}{2} a - \frac{1406571487177}{7315127527864}$, $\frac{1}{716882497730672} a^{19} - \frac{1}{102411785390096} a^{18} + \frac{44143578875}{102411785390096} a^{17} - \frac{27653993361}{102411785390096} a^{16} - \frac{2607630974969}{716882497730672} a^{15} - \frac{1049932788963}{102411785390096} a^{14} + \frac{115910853945983}{716882497730672} a^{13} - \frac{15468525967235}{102411785390096} a^{12} - \frac{9272582377727}{51205892695048} a^{11} - \frac{4585634870619}{51205892695048} a^{10} - \frac{15362725247479}{89610312216334} a^{9} + \frac{9272432264203}{25602946347524} a^{8} - \frac{3408991892097}{7315127527864} a^{7} + \frac{7936578981711}{51205892695048} a^{6} - \frac{69969645585955}{179220624432668} a^{5} - \frac{2501098216928}{6400736586881} a^{4} + \frac{269780787786329}{716882497730672} a^{3} - \frac{39354270658613}{102411785390096} a^{2} - \frac{13645342543411}{102411785390096} a - \frac{4079774158721}{14630255055728}$

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:  $14$
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:  \( 3713936628.33 \) (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 $20$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{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}$
13Data not computed