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Properties

Label 18.0.23220382749...7771.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{44}\cdot 11^{9}$
Root discriminant $48.64$
Ramified primes $3, 11$
Class number $1791$ (GRH)
Class group $[1791]$ (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![992091, -860841, 1272483, -933768, 814050, -518733, 351249, -193914, 108099, -51962, 24552, -10422, 4272, -1512, 504, -156, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 504*x^14 - 1512*x^13 + 4272*x^12 - 10422*x^11 + 24552*x^10 - 51962*x^9 + 108099*x^8 - 193914*x^7 + 351249*x^6 - 518733*x^5 + 814050*x^4 - 933768*x^3 + 1272483*x^2 - 860841*x + 992091)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 504*x^14 - 1512*x^13 + 4272*x^12 - 10422*x^11 + 24552*x^10 - 51962*x^9 + 108099*x^8 - 193914*x^7 + 351249*x^6 - 518733*x^5 + 814050*x^4 - 933768*x^3 + 1272483*x^2 - 860841*x + 992091, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 156 x^{15} + 504 x^{14} - 1512 x^{13} + 4272 x^{12} - 10422 x^{11} + 24552 x^{10} - 51962 x^{9} + 108099 x^{8} - 193914 x^{7} + 351249 x^{6} - 518733 x^{5} + 814050 x^{4} - 933768 x^{3} + 1272483 x^{2} - 860841 x + 992091 \)

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:  \(-2322038274967832964613417227771=-\,3^{44}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.64$
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:  $3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(297=3^{3}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{297}(1,·)$, $\chi_{297}(67,·)$, $\chi_{297}(133,·)$, $\chi_{297}(199,·)$, $\chi_{297}(265,·)$, $\chi_{297}(10,·)$, $\chi_{297}(76,·)$, $\chi_{297}(142,·)$, $\chi_{297}(208,·)$, $\chi_{297}(274,·)$, $\chi_{297}(34,·)$, $\chi_{297}(100,·)$, $\chi_{297}(166,·)$, $\chi_{297}(232,·)$, $\chi_{297}(43,·)$, $\chi_{297}(109,·)$, $\chi_{297}(175,·)$, $\chi_{297}(241,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{53} a^{16} - \frac{17}{53} a^{15} - \frac{10}{53} a^{14} - \frac{9}{53} a^{13} - \frac{5}{53} a^{12} - \frac{18}{53} a^{11} + \frac{18}{53} a^{10} - \frac{26}{53} a^{9} + \frac{16}{53} a^{8} - \frac{7}{53} a^{7} + \frac{25}{53} a^{5} - \frac{23}{53} a^{4} - \frac{2}{53} a^{3} - \frac{20}{53} a^{2} - \frac{2}{53} a + \frac{26}{53}$, $\frac{1}{34169207459977467101463968260835724840893} a^{17} + \frac{74702070290591842567599113916041164529}{34169207459977467101463968260835724840893} a^{16} + \frac{16934024691184908370691142181567855006393}{34169207459977467101463968260835724840893} a^{15} - \frac{10169241116171378359967406796097855654359}{34169207459977467101463968260835724840893} a^{14} + \frac{13095148955058156249095336276127644299015}{34169207459977467101463968260835724840893} a^{13} - \frac{8659869122806934916787980423875743009374}{34169207459977467101463968260835724840893} a^{12} + \frac{1524511619608715687702695389250937001269}{34169207459977467101463968260835724840893} a^{11} - \frac{16265901593002288500908810232761163449303}{34169207459977467101463968260835724840893} a^{10} - \frac{7360198747608851375774859101163775870168}{34169207459977467101463968260835724840893} a^{9} - \frac{12946149880015593877331645420924168540782}{34169207459977467101463968260835724840893} a^{8} - \frac{16380810862811733609277167453909955413879}{34169207459977467101463968260835724840893} a^{7} - \frac{10612256592505140489840392793186177357522}{34169207459977467101463968260835724840893} a^{6} + \frac{6438487699785836555437597172861068390651}{34169207459977467101463968260835724840893} a^{5} + \frac{10087580776169055905295840101726134117781}{34169207459977467101463968260835724840893} a^{4} - \frac{14364899010791172871648139849802121945442}{34169207459977467101463968260835724840893} a^{3} - \frac{9167038593120350835774013135868593804859}{34169207459977467101463968260835724840893} a^{2} + \frac{11141308888863326655833815249761597613515}{34169207459977467101463968260835724840893} a - \frac{7666495068171568474937539195213172302344}{34169207459977467101463968260835724840893}$

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

Class group and class number

$C_{1791}$, which has order $1791$ (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:  \( 40934.0329443 \) (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

\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{9})^+\), 6.0.8732691.1, \(\Q(\zeta_{27})^+\)

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 $18$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ $18$ R $18$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ $18$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$3$3.9.22.8$x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$$9$$1$$22$$C_9$$[2, 3]$
3.9.22.8$x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$$9$$1$$22$$C_9$$[2, 3]$
11Data not computed