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Properties

Label 20.0.32473988377...0464.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{10}\cdot 11^{17}\cdot 89^{4}$
Root discriminant $26.64$
Ramified primes $2, 11, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T427

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7019, 1453, -6236, -5691, 13836, 3978, -2167, -6244, 1775, -584, 1737, 480, -842, -164, 278, -39, -23, 15, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 15*x^17 - 23*x^16 - 39*x^15 + 278*x^14 - 164*x^13 - 842*x^12 + 480*x^11 + 1737*x^10 - 584*x^9 + 1775*x^8 - 6244*x^7 - 2167*x^6 + 3978*x^5 + 13836*x^4 - 5691*x^3 - 6236*x^2 + 1453*x + 7019)
 
gp: K = bnfinit(x^20 - 3*x^19 + 15*x^17 - 23*x^16 - 39*x^15 + 278*x^14 - 164*x^13 - 842*x^12 + 480*x^11 + 1737*x^10 - 584*x^9 + 1775*x^8 - 6244*x^7 - 2167*x^6 + 3978*x^5 + 13836*x^4 - 5691*x^3 - 6236*x^2 + 1453*x + 7019, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 15 x^{17} - 23 x^{16} - 39 x^{15} + 278 x^{14} - 164 x^{13} - 842 x^{12} + 480 x^{11} + 1737 x^{10} - 584 x^{9} + 1775 x^{8} - 6244 x^{7} - 2167 x^{6} + 3978 x^{5} + 13836 x^{4} - 5691 x^{3} - 6236 x^{2} + 1453 x + 7019 \)

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:  \(32473988377432635504517950464=2^{10}\cdot 11^{17}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.64$
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:  $2, 11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{883275517522730318228774855151356334166002589} a^{19} + \frac{162127924880105914409373161507520523897033120}{883275517522730318228774855151356334166002589} a^{18} - \frac{347013657594900455863635710152777226095897582}{883275517522730318228774855151356334166002589} a^{17} + \frac{345673886088516097439901973724356957012351660}{883275517522730318228774855151356334166002589} a^{16} + \frac{131925548172391219680818291230690978542197240}{883275517522730318228774855151356334166002589} a^{15} - \frac{215701541521403856767247530671025294353662425}{883275517522730318228774855151356334166002589} a^{14} + \frac{145726935739848874219618279878766061524466300}{883275517522730318228774855151356334166002589} a^{13} + \frac{260293540577340486234577609802946701921174282}{883275517522730318228774855151356334166002589} a^{12} + \frac{418830894827521648902633246847006640071868840}{883275517522730318228774855151356334166002589} a^{11} - \frac{334139550479254767364776075719106228714760136}{883275517522730318228774855151356334166002589} a^{10} - \frac{324918728687111323812789869025884683130928056}{883275517522730318228774855151356334166002589} a^{9} - \frac{399196902432106500667354114732982778900358333}{883275517522730318228774855151356334166002589} a^{8} + \frac{106407695059336459810967381846453733916590192}{883275517522730318228774855151356334166002589} a^{7} - \frac{314531243797466903045267898894286783837490681}{883275517522730318228774855151356334166002589} a^{6} - \frac{129094608340818766367944957365381084419600356}{883275517522730318228774855151356334166002589} a^{5} - \frac{295798732135286974608674954337995663491906636}{883275517522730318228774855151356334166002589} a^{4} - \frac{292384654759721554301427982591583903086051136}{883275517522730318228774855151356334166002589} a^{3} - \frac{17786993568855231816713581991348661474152305}{883275517522730318228774855151356334166002589} a^{2} - \frac{364535543405847467511689825153005091150790745}{883275517522730318228774855151356334166002589} a - \frac{188224978876121928447762215729587175536172378}{883275517522730318228774855151356334166002589}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 471583.031519 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T427:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n427 are not computed
Character table for t20n427 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.2.19077940409.1

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 $20$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$