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Properties

Label 20.8.48960645842...8608.2
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 13^{14}\cdot 17^{9}$
Root discriminant $43.10$
Ramified primes $2, 13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T803

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-452, -104, 12252, -33056, 29868, 20424, -87016, 121188, -112192, 79288, -44968, 21568, -8972, 3224, -1286, 706, -336, 96, -8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 8*x^18 + 96*x^17 - 336*x^16 + 706*x^15 - 1286*x^14 + 3224*x^13 - 8972*x^12 + 21568*x^11 - 44968*x^10 + 79288*x^9 - 112192*x^8 + 121188*x^7 - 87016*x^6 + 20424*x^5 + 29868*x^4 - 33056*x^3 + 12252*x^2 - 104*x - 452)
 
gp: K = bnfinit(x^20 - 4*x^19 - 8*x^18 + 96*x^17 - 336*x^16 + 706*x^15 - 1286*x^14 + 3224*x^13 - 8972*x^12 + 21568*x^11 - 44968*x^10 + 79288*x^9 - 112192*x^8 + 121188*x^7 - 87016*x^6 + 20424*x^5 + 29868*x^4 - 33056*x^3 + 12252*x^2 - 104*x - 452, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 8 x^{18} + 96 x^{17} - 336 x^{16} + 706 x^{15} - 1286 x^{14} + 3224 x^{13} - 8972 x^{12} + 21568 x^{11} - 44968 x^{10} + 79288 x^{9} - 112192 x^{8} + 121188 x^{7} - 87016 x^{6} + 20424 x^{5} + 29868 x^{4} - 33056 x^{3} + 12252 x^{2} - 104 x - 452 \)

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:  \(489606458428108592552566464708608=2^{20}\cdot 13^{14}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.10$
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:  $2, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{17} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7}$, $\frac{1}{284} a^{18} + \frac{19}{284} a^{17} - \frac{11}{284} a^{16} - \frac{17}{71} a^{15} - \frac{21}{142} a^{14} - \frac{9}{142} a^{13} + \frac{6}{71} a^{12} + \frac{13}{71} a^{11} - \frac{9}{142} a^{10} + \frac{30}{71} a^{9} + \frac{19}{71} a^{8} + \frac{30}{71} a^{7} + \frac{61}{142} a^{6} + \frac{13}{71} a^{5} - \frac{14}{71} a^{4} - \frac{13}{71} a^{3} - \frac{20}{71} a^{2} - \frac{22}{71} a - \frac{3}{71}$, $\frac{1}{1679751451758964283194329208501660} a^{19} - \frac{595717678629562807119693870458}{419937862939741070798582302125415} a^{18} + \frac{142281854197877236399428890500623}{1679751451758964283194329208501660} a^{17} - \frac{24497780507890929848551668880959}{839875725879482141597164604250830} a^{16} + \frac{139335639926179969112642931452059}{839875725879482141597164604250830} a^{15} - \frac{89152197753096105775400696394447}{419937862939741070798582302125415} a^{14} - \frac{12678971341560243364465741472031}{839875725879482141597164604250830} a^{13} - \frac{21087532810873804005517992887839}{167975145175896428319432920850166} a^{12} - \frac{98540044960378481796712864618768}{419937862939741070798582302125415} a^{11} + \frac{91803198006281073996010959731036}{419937862939741070798582302125415} a^{10} - \frac{30343523659137405062628644004287}{167975145175896428319432920850166} a^{9} - \frac{334991838624206810251524811323261}{839875725879482141597164604250830} a^{8} - \frac{3079250534151265676248963416491}{8154133260965846034923928196610} a^{7} + \frac{7719410090073377304609768827134}{419937862939741070798582302125415} a^{6} - \frac{96237176347802965386754721536336}{419937862939741070798582302125415} a^{5} + \frac{43147754384810887835020056416564}{419937862939741070798582302125415} a^{4} + \frac{16509970998968259962334639283351}{83987572587948214159716460425083} a^{3} + \frac{139804734776552568984548725303261}{419937862939741070798582302125415} a^{2} + \frac{32861690382015926680834610079151}{83987572587948214159716460425083} a - \frac{9876459115615360610284501192966}{419937862939741070798582302125415}$

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:  \( 939462904.818 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T803:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 126 conjugacy class representatives for t20n803 are not computed
Character table for t20n803 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.5.10158928.1, 10.10.1341649635419392.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.12.10.2$x^{12} + 39 x^{6} + 676$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.6.5.2$x^{6} + 51$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$