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Properties

Label 20.14.9584766433...8672.1
Degree $20$
Signature $[14, 3]$
Discriminant $-\,2^{28}\cdot 13^{15}\cdot 17^{8}$
Root discriminant $56.12$
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![299, -3224, 5096, 33124, -120159, 85254, 138490, -195946, -19932, 118742, -33804, -20006, 15238, -3770, -1246, 1262, -303, -26, 36, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 36*x^18 - 26*x^17 - 303*x^16 + 1262*x^15 - 1246*x^14 - 3770*x^13 + 15238*x^12 - 20006*x^11 - 33804*x^10 + 118742*x^9 - 19932*x^8 - 195946*x^7 + 138490*x^6 + 85254*x^5 - 120159*x^4 + 33124*x^3 + 5096*x^2 - 3224*x + 299)
 
gp: K = bnfinit(x^20 - 10*x^19 + 36*x^18 - 26*x^17 - 303*x^16 + 1262*x^15 - 1246*x^14 - 3770*x^13 + 15238*x^12 - 20006*x^11 - 33804*x^10 + 118742*x^9 - 19932*x^8 - 195946*x^7 + 138490*x^6 + 85254*x^5 - 120159*x^4 + 33124*x^3 + 5096*x^2 - 3224*x + 299, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 36 x^{18} - 26 x^{17} - 303 x^{16} + 1262 x^{15} - 1246 x^{14} - 3770 x^{13} + 15238 x^{12} - 20006 x^{11} - 33804 x^{10} + 118742 x^{9} - 19932 x^{8} - 195946 x^{7} + 138490 x^{6} + 85254 x^{5} - 120159 x^{4} + 33124 x^{3} + 5096 x^{2} - 3224 x + 299 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-95847664332279140942055364385308672=-\,2^{28}\cdot 13^{15}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.12$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a$, $\frac{1}{182} a^{18} - \frac{2}{13} a^{17} + \frac{15}{91} a^{16} - \frac{27}{182} a^{15} - \frac{11}{182} a^{14} + \frac{33}{182} a^{13} + \frac{3}{14} a^{12} - \frac{4}{91} a^{11} + \frac{3}{13} a^{10} + \frac{31}{91} a^{9} + \frac{40}{91} a^{8} + \frac{51}{182} a^{7} - \frac{43}{182} a^{6} + \frac{3}{14} a^{5} + \frac{1}{14} a^{4} - \frac{1}{7} a^{3} - \frac{1}{14} a^{2} + \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{3019408367592222468113414695202530678} a^{19} - \frac{754353656539251349604928527806917}{1509704183796111234056707347601265339} a^{18} + \frac{125138698432653867692860516987026309}{1509704183796111234056707347601265339} a^{17} + \frac{76187267426099406093619509831150}{1509704183796111234056707347601265339} a^{16} + \frac{168754242719399748056969356387709549}{3019408367592222468113414695202530678} a^{15} + \frac{1458416007115029977324261973441082}{116131091061239325696669795969328103} a^{14} + \frac{43996802709799715873500746111996867}{431344052513174638301916385028932954} a^{13} + \frac{47615409841167669522709751377779496}{215672026256587319150958192514466477} a^{12} + \frac{259146244242016924359906035304449847}{3019408367592222468113414695202530678} a^{11} + \frac{585548627353405655451688632062974009}{3019408367592222468113414695202530678} a^{10} + \frac{606860866803461115663505384855876243}{3019408367592222468113414695202530678} a^{9} - \frac{145993259229621298311592776274407895}{3019408367592222468113414695202530678} a^{8} - \frac{887275060602901412280627810055830569}{3019408367592222468113414695202530678} a^{7} - \frac{604908785176705488519973220340868913}{1509704183796111234056707347601265339} a^{6} - \frac{7357837408296992719757999480167341}{33180311731782664484762798848379458} a^{5} - \frac{3422854967979335466736903560245142}{16590155865891332242381399424189729} a^{4} + \frac{17455783593098632267013526652146585}{116131091061239325696669795969328103} a^{3} - \frac{15604938044868342104244869360532517}{33180311731782664484762798848379458} a^{2} - \frac{1359674141993467684424486069522029}{232262182122478651393339591938656206} a - \frac{67756466019684140530323177459569583}{232262182122478651393339591938656206}$

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:  $16$
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:  \( 51933061040.3 \) (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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
$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}$
17.2.0.1$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}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$