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Properties

Label 18.0.68012976511...4571.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,11^{9}\cdot 19^{16}$
Root discriminant $45.43$
Ramified primes $11, 19$
Class number $1084$ (GRH)
Class group $[2, 542]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![713641, -453934, 833369, -511304, 517517, -293668, 216508, -111391, 66392, -30674, 15226, -6227, 2693, -957, 331, -98, 31, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 31*x^16 - 98*x^15 + 331*x^14 - 957*x^13 + 2693*x^12 - 6227*x^11 + 15226*x^10 - 30674*x^9 + 66392*x^8 - 111391*x^7 + 216508*x^6 - 293668*x^5 + 517517*x^4 - 511304*x^3 + 833369*x^2 - 453934*x + 713641)
 
gp: K = bnfinit(x^18 - 7*x^17 + 31*x^16 - 98*x^15 + 331*x^14 - 957*x^13 + 2693*x^12 - 6227*x^11 + 15226*x^10 - 30674*x^9 + 66392*x^8 - 111391*x^7 + 216508*x^6 - 293668*x^5 + 517517*x^4 - 511304*x^3 + 833369*x^2 - 453934*x + 713641, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 31 x^{16} - 98 x^{15} + 331 x^{14} - 957 x^{13} + 2693 x^{12} - 6227 x^{11} + 15226 x^{10} - 30674 x^{9} + 66392 x^{8} - 111391 x^{7} + 216508 x^{6} - 293668 x^{5} + 517517 x^{4} - 511304 x^{3} + 833369 x^{2} - 453934 x + 713641 \)

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:  \(-680129765110548404696137774571=-\,11^{9}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.43$
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:  $11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(209=11\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{209}(1,·)$, $\chi_{209}(131,·)$, $\chi_{209}(197,·)$, $\chi_{209}(199,·)$, $\chi_{209}(23,·)$, $\chi_{209}(142,·)$, $\chi_{209}(144,·)$, $\chi_{209}(87,·)$, $\chi_{209}(153,·)$, $\chi_{209}(111,·)$, $\chi_{209}(100,·)$, $\chi_{209}(43,·)$, $\chi_{209}(45,·)$, $\chi_{209}(175,·)$, $\chi_{209}(177,·)$, $\chi_{209}(54,·)$, $\chi_{209}(120,·)$, $\chi_{209}(188,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{37} a^{16} - \frac{12}{37} a^{15} + \frac{8}{37} a^{14} + \frac{7}{37} a^{13} + \frac{2}{37} a^{12} + \frac{6}{37} a^{11} + \frac{18}{37} a^{10} - \frac{7}{37} a^{9} + \frac{3}{37} a^{8} + \frac{10}{37} a^{7} + \frac{11}{37} a^{6} - \frac{18}{37} a^{5} + \frac{12}{37} a^{4} - \frac{8}{37} a^{3} + \frac{4}{37} a^{2} + \frac{14}{37} a - \frac{14}{37}$, $\frac{1}{28457777823908436935405793946214340663563} a^{17} - \frac{237595025782776872540170486702865567274}{28457777823908436935405793946214340663563} a^{16} - \frac{12258887164629929159459016023621380345437}{28457777823908436935405793946214340663563} a^{15} + \frac{8280097195585768469456862774160164774581}{28457777823908436935405793946214340663563} a^{14} + \frac{11862586698276224228301778034099339949891}{28457777823908436935405793946214340663563} a^{13} - \frac{5646246221193397306578180503232518117175}{28457777823908436935405793946214340663563} a^{12} - \frac{12673440368371598293107464799259115642299}{28457777823908436935405793946214340663563} a^{11} - \frac{10349552141364950614120209563522760158265}{28457777823908436935405793946214340663563} a^{10} - \frac{9610973594250407378453718903253159313288}{28457777823908436935405793946214340663563} a^{9} - \frac{5960143297318532701612903997057590367307}{28457777823908436935405793946214340663563} a^{8} + \frac{9929843569041625984574128413123656095083}{28457777823908436935405793946214340663563} a^{7} + \frac{7623034380104726107420892426846581433356}{28457777823908436935405793946214340663563} a^{6} - \frac{9604348876139736354561538776239225059868}{28457777823908436935405793946214340663563} a^{5} + \frac{729506287298409805420002919065689042259}{28457777823908436935405793946214340663563} a^{4} + \frac{4834831343607227562401662999367992661778}{28457777823908436935405793946214340663563} a^{3} + \frac{10251310127855850191545705554573076775938}{28457777823908436935405793946214340663563} a^{2} - \frac{11846586489147974672463398730583717574441}{28457777823908436935405793946214340663563} a - \frac{12896555728830716897942688844386021206202}{28457777823908436935405793946214340663563}$

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

Class group and class number

$C_{2}\times C_{542}$, which has order $1084$ (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:  \( 22305.8950792 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-11}) \), 3.3.361.1, 6.0.173457251.1, \(\Q(\zeta_{19})^+\)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $18$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R $18$ $18$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ $18$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19Data not computed