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Properties

Label 18.0.29902909752...4247.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 71^{3}\cdot 113^{3}\cdot 64679^{2}$
Root discriminant $56.06$
Ramified primes $7, 71, 113, 64679$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group 18T472

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14591569, 2010611, 16839843, 2593860, 8662708, 1204439, 2635061, 265934, 522337, 25446, 71045, -380, 7100, -279, 567, -13, 34, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 34*x^16 - 13*x^15 + 567*x^14 - 279*x^13 + 7100*x^12 - 380*x^11 + 71045*x^10 + 25446*x^9 + 522337*x^8 + 265934*x^7 + 2635061*x^6 + 1204439*x^5 + 8662708*x^4 + 2593860*x^3 + 16839843*x^2 + 2010611*x + 14591569)
 
gp: K = bnfinit(x^18 + 34*x^16 - 13*x^15 + 567*x^14 - 279*x^13 + 7100*x^12 - 380*x^11 + 71045*x^10 + 25446*x^9 + 522337*x^8 + 265934*x^7 + 2635061*x^6 + 1204439*x^5 + 8662708*x^4 + 2593860*x^3 + 16839843*x^2 + 2010611*x + 14591569, 1)
 

Normalized defining polynomial

\( x^{18} + 34 x^{16} - 13 x^{15} + 567 x^{14} - 279 x^{13} + 7100 x^{12} - 380 x^{11} + 71045 x^{10} + 25446 x^{9} + 522337 x^{8} + 265934 x^{7} + 2635061 x^{6} + 1204439 x^{5} + 8662708 x^{4} + 2593860 x^{3} + 16839843 x^{2} + 2010611 x + 14591569 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-29902909752142884167850994584247=-\,7^{12}\cdot 71^{3}\cdot 113^{3}\cdot 64679^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.06$
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:  $7, 71, 113, 64679$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13} a^{16} + \frac{6}{13} a^{15} + \frac{2}{13} a^{14} - \frac{6}{13} a^{13} + \frac{5}{13} a^{12} + \frac{3}{13} a^{11} + \frac{5}{13} a^{10} + \frac{5}{13} a^{9} + \frac{2}{13} a^{8} + \frac{2}{13} a^{7} + \frac{3}{13} a^{6} + \frac{5}{13} a^{5} - \frac{5}{13} a^{4} - \frac{4}{13} a^{3} + \frac{6}{13} a^{2} + \frac{5}{13} a + \frac{6}{13}$, $\frac{1}{1326317894176976991857346587814932241960662339586161} a^{17} - \frac{44347939846466037440029184559728325444290688262887}{1326317894176976991857346587814932241960662339586161} a^{16} + \frac{101444541248024860273047589673255756310493005419914}{1326317894176976991857346587814932241960662339586161} a^{15} + \frac{657562511100952203654057293799708654531630623843514}{1326317894176976991857346587814932241960662339586161} a^{14} + \frac{653720337755533712348497681655152358207149278508176}{1326317894176976991857346587814932241960662339586161} a^{13} - \frac{26976140605366582646517606519054513367867510378791}{1326317894176976991857346587814932241960662339586161} a^{12} - \frac{360132136927341189318497660306001261118439406945334}{1326317894176976991857346587814932241960662339586161} a^{11} + \frac{75526593184084366623625178491646520431579838237408}{1326317894176976991857346587814932241960662339586161} a^{10} - \frac{139414826750450216068824187751542737832585554238754}{1326317894176976991857346587814932241960662339586161} a^{9} - \frac{419675304199308078822374642110754981865280034090091}{1326317894176976991857346587814932241960662339586161} a^{8} - \frac{491798537727898522693757616360189400909315828317460}{1326317894176976991857346587814932241960662339586161} a^{7} - \frac{515097070519025770618903807838922792269100888389884}{1326317894176976991857346587814932241960662339586161} a^{6} + \frac{16277726759629678122732366286207692038732366331360}{102024453398228999373642045216533249381589410737397} a^{5} + \frac{480890402528269658039195630119101769411355714988329}{1326317894176976991857346587814932241960662339586161} a^{4} - \frac{270073059914407051901356468463242418242981713813375}{1326317894176976991857346587814932241960662339586161} a^{3} - \frac{323909852612150596005021714648914005708333677166652}{1326317894176976991857346587814932241960662339586161} a^{2} + \frac{2903307302726188303704302907886009146283441031810}{1326317894176976991857346587814932241960662339586161} a + \frac{477976221522929793182553146754439176320362374759500}{1326317894176976991857346587814932241960662339586161}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  $8$
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:  \( 26315478.7454 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T472:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5184
The 88 conjugacy class representatives for t18n472 are not computed
Character table for t18n472 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.0.19263223.1, 9.7.7609419671.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ $18$ R $18$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
7Data not computed
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
71.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
71.6.3.1$x^{6} - 142 x^{4} + 5041 x^{2} - 1431644$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$113$$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.6.3.1$x^{6} - 226 x^{4} + 12769 x^{2} - 36072425$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
64679Data not computed