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Properties

Label 20.8.60773814096...0625.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{17}\cdot 6029^{5}$
Root discriminant $34.61$
Ramified primes $5, 6029$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T797

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, 5, -910, -2325, -1685, 1040, 1625, 935, 1795, 165, -1120, -85, -24, -146, -58, 42, 45, -12, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 2*x^18 - 12*x^17 + 45*x^16 + 42*x^15 - 58*x^14 - 146*x^13 - 24*x^12 - 85*x^11 - 1120*x^10 + 165*x^9 + 1795*x^8 + 935*x^7 + 1625*x^6 + 1040*x^5 - 1685*x^4 - 2325*x^3 - 910*x^2 + 5*x + 5)
 
gp: K = bnfinit(x^20 - 4*x^19 + 2*x^18 - 12*x^17 + 45*x^16 + 42*x^15 - 58*x^14 - 146*x^13 - 24*x^12 - 85*x^11 - 1120*x^10 + 165*x^9 + 1795*x^8 + 935*x^7 + 1625*x^6 + 1040*x^5 - 1685*x^4 - 2325*x^3 - 910*x^2 + 5*x + 5, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 2 x^{18} - 12 x^{17} + 45 x^{16} + 42 x^{15} - 58 x^{14} - 146 x^{13} - 24 x^{12} - 85 x^{11} - 1120 x^{10} + 165 x^{9} + 1795 x^{8} + 935 x^{7} + 1625 x^{6} + 1040 x^{5} - 1685 x^{4} - 2325 x^{3} - 910 x^{2} + 5 x + 5 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6077381409667160910797119140625=5^{17}\cdot 6029^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.61$
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:  $5, 6029$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} - \frac{2}{5} a^{15} - \frac{1}{5} a^{14} + \frac{2}{5} a^{13} + \frac{1}{5} a^{12}$, $\frac{1}{5} a^{17} + \frac{2}{5} a^{12}$, $\frac{1}{5} a^{18} + \frac{2}{5} a^{13}$, $\frac{1}{732434671021074474369094132773925} a^{19} + \frac{40256822066494539964913120563339}{732434671021074474369094132773925} a^{18} - \frac{30784264441734313506294791895541}{732434671021074474369094132773925} a^{17} + \frac{11397606811274608767792529651329}{146486934204214894873818826554785} a^{16} - \frac{66306767920312072173918680931946}{146486934204214894873818826554785} a^{15} + \frac{333889951651990843629312278896022}{732434671021074474369094132773925} a^{14} - \frac{18022299743263119799171206964202}{732434671021074474369094132773925} a^{13} - \frac{355581208170864824266547401122992}{732434671021074474369094132773925} a^{12} + \frac{3951260055915831888901486245118}{29297386840842978974763765310957} a^{11} + \frac{12194990929346579778905506653968}{146486934204214894873818826554785} a^{10} - \frac{8262559469552614081052952197645}{29297386840842978974763765310957} a^{9} + \frac{20345947292612479406593304218283}{146486934204214894873818826554785} a^{8} - \frac{16153525308643204149503605250787}{146486934204214894873818826554785} a^{7} + \frac{71537856807381926073784244816996}{146486934204214894873818826554785} a^{6} - \frac{55802359473424655280742079086047}{146486934204214894873818826554785} a^{5} - \frac{58513679919852957470341484734788}{146486934204214894873818826554785} a^{4} + \frac{63840224064599145282078636285884}{146486934204214894873818826554785} a^{3} + \frac{44430384107773071554121990714207}{146486934204214894873818826554785} a^{2} + \frac{50678608004918591477389538896329}{146486934204214894873818826554785} a + \frac{7393704109881600553114062192188}{146486934204214894873818826554785}$

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:  $13$
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:  \( 58496924.2793 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T797:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n797 are not computed
Character table for t20n797 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.11.2$x^{12} - 20$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$
6029Data not computed