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Properties

Label 20.8.21070818384...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{32}\cdot 5^{10}\cdot 3469^{5}$
Root discriminant $52.02$
Ramified primes $2, 5, 3469$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T756

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11695, -68050, -54659, 128112, 145545, 12894, 13398, 16976, 261, -24676, -31791, -1542, 6193, 1818, -654, -270, -37, 54, 3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 3*x^18 + 54*x^17 - 37*x^16 - 270*x^15 - 654*x^14 + 1818*x^13 + 6193*x^12 - 1542*x^11 - 31791*x^10 - 24676*x^9 + 261*x^8 + 16976*x^7 + 13398*x^6 + 12894*x^5 + 145545*x^4 + 128112*x^3 - 54659*x^2 - 68050*x - 11695)
 
gp: K = bnfinit(x^20 - 6*x^19 + 3*x^18 + 54*x^17 - 37*x^16 - 270*x^15 - 654*x^14 + 1818*x^13 + 6193*x^12 - 1542*x^11 - 31791*x^10 - 24676*x^9 + 261*x^8 + 16976*x^7 + 13398*x^6 + 12894*x^5 + 145545*x^4 + 128112*x^3 - 54659*x^2 - 68050*x - 11695, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 3 x^{18} + 54 x^{17} - 37 x^{16} - 270 x^{15} - 654 x^{14} + 1818 x^{13} + 6193 x^{12} - 1542 x^{11} - 31791 x^{10} - 24676 x^{9} + 261 x^{8} + 16976 x^{7} + 13398 x^{6} + 12894 x^{5} + 145545 x^{4} + 128112 x^{3} - 54659 x^{2} - 68050 x - 11695 \)

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:  \(21070818384983657241640960000000000=2^{32}\cdot 5^{10}\cdot 3469^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.02$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{15} a^{17} + \frac{2}{15} a^{16} + \frac{1}{15} a^{15} + \frac{7}{15} a^{14} - \frac{7}{15} a^{13} - \frac{1}{15} a^{12} - \frac{4}{15} a^{11} - \frac{1}{5} a^{10} - \frac{1}{3} a^{9} - \frac{2}{15} a^{8} - \frac{1}{15} a^{5} + \frac{1}{5} a^{4} + \frac{1}{3} a^{3} - \frac{2}{5} a^{2} - \frac{1}{3}$, $\frac{1}{75} a^{18} + \frac{1}{75} a^{17} + \frac{4}{75} a^{16} + \frac{1}{75} a^{15} - \frac{29}{75} a^{14} + \frac{7}{25} a^{13} - \frac{11}{25} a^{12} + \frac{12}{25} a^{11} - \frac{32}{75} a^{10} - \frac{7}{75} a^{9} + \frac{2}{75} a^{8} + \frac{1}{15} a^{7} - \frac{16}{75} a^{6} - \frac{16}{75} a^{5} + \frac{32}{75} a^{4} - \frac{11}{75} a^{3} + \frac{1}{75} a^{2} + \frac{2}{15} a - \frac{4}{15}$, $\frac{1}{208730501751200541815422991260712254812170591951696625} a^{19} - \frac{219593687906113724506162634284488184487685459789436}{41746100350240108363084598252142450962434118390339325} a^{18} + \frac{1319813780432465227630221986446982290640469889828198}{208730501751200541815422991260712254812170591951696625} a^{17} - \frac{3276450282758309089831750558763079732033225796400466}{69576833917066847271807663753570751604056863983898875} a^{16} + \frac{2292629575443221246390853597564855206697468771925156}{13915366783413369454361532750714150320811372796779775} a^{15} - \frac{4690798366118086961722417364492327067039170091196936}{41746100350240108363084598252142450962434118390339325} a^{14} - \frac{75433713344704547473143161226563134174711327576267459}{208730501751200541815422991260712254812170591951696625} a^{13} + \frac{6485127120534813028489322703898610906365628634032434}{208730501751200541815422991260712254812170591951696625} a^{12} - \frac{65715133665967384152738766540497884111637993232066948}{208730501751200541815422991260712254812170591951696625} a^{11} + \frac{13092077380639529109786494136769788213573494612176307}{41746100350240108363084598252142450962434118390339325} a^{10} + \frac{33526180949444097898433887167691720759288560868420244}{208730501751200541815422991260712254812170591951696625} a^{9} + \frac{51615576180622793595737638250474992036088888627336618}{208730501751200541815422991260712254812170591951696625} a^{8} + \frac{31999970599312855661675216669217140789639316508017979}{208730501751200541815422991260712254812170591951696625} a^{7} - \frac{4573637543469387203068645494713122705887487929220308}{13915366783413369454361532750714150320811372796779775} a^{6} - \frac{18711209567879909510179821544227169261970089103902972}{208730501751200541815422991260712254812170591951696625} a^{5} + \frac{36046350556193142331638949219254647210098673283049147}{208730501751200541815422991260712254812170591951696625} a^{4} - \frac{18105908820134831331657280013448966977390682497778908}{208730501751200541815422991260712254812170591951696625} a^{3} - \frac{42653830096748251939117702236133096059782552340634396}{208730501751200541815422991260712254812170591951696625} a^{2} + \frac{252957786170039569892474603576718276780039112472933}{13915366783413369454361532750714150320811372796779775} a - \frac{1573378104496705790321693520004037486130859542309562}{13915366783413369454361532750714150320811372796779775}$

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:  \( 7344599258.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T756:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n756 are not computed
Character table for t20n756 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3469Data not computed