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Properties

Label 18.0.21730892405...4656.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{45}\cdot 19^{17}$
Root discriminant $711.31$
Ramified primes $2, 3, 19$
Class number $1583299512$ (GRH)
Class group $[2, 791649756]$ (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![188728521216, 0, 829422602496, 0, 444387949056, 0, 88486683264, 0, 8467405632, 0, 426862512, 0, 11366256, 0, 145692, 0, 684, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 684*x^16 + 145692*x^14 + 11366256*x^12 + 426862512*x^10 + 8467405632*x^8 + 88486683264*x^6 + 444387949056*x^4 + 829422602496*x^2 + 188728521216)
 
gp: K = bnfinit(x^18 + 684*x^16 + 145692*x^14 + 11366256*x^12 + 426862512*x^10 + 8467405632*x^8 + 88486683264*x^6 + 444387949056*x^4 + 829422602496*x^2 + 188728521216, 1)
 

Normalized defining polynomial

\( x^{18} + 684 x^{16} + 145692 x^{14} + 11366256 x^{12} + 426862512 x^{10} + 8467405632 x^{8} + 88486683264 x^{6} + 444387949056 x^{4} + 829422602496 x^{2} + 188728521216 \)

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:  \(-2173089240515381294985625698929389621300732316614656=-\,2^{27}\cdot 3^{45}\cdot 19^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $711.31$
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:  $2, 3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4104=2^{3}\cdot 3^{3}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4104}(1,·)$, $\chi_{4104}(2051,·)$, $\chi_{4104}(1537,·)$, $\chi_{4104}(1283,·)$, $\chi_{4104}(2569,·)$, $\chi_{4104}(1547,·)$, $\chi_{4104}(1475,·)$, $\chi_{4104}(529,·)$, $\chi_{4104}(3587,·)$, $\chi_{4104}(1667,·)$, $\chi_{4104}(481,·)$, $\chi_{4104}(1571,·)$, $\chi_{4104}(515,·)$, $\chi_{4104}(385,·)$, $\chi_{4104}(1523,·)$, $\chi_{4104}(577,·)$, $\chi_{4104}(505,·)$, $\chi_{4104}(769,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{1856} a^{12} - \frac{13}{928} a^{10} - \frac{1}{232} a^{8} + \frac{3}{58} a^{6} + \frac{1}{58} a^{4} - \frac{13}{58} a^{2} - \frac{1}{29}$, $\frac{1}{1856} a^{13} - \frac{13}{928} a^{11} - \frac{1}{232} a^{9} + \frac{3}{58} a^{7} + \frac{1}{58} a^{5} - \frac{13}{58} a^{3} - \frac{1}{29} a$, $\frac{1}{3712} a^{14} + \frac{3}{928} a^{10} - \frac{7}{232} a^{8} + \frac{13}{232} a^{6} + \frac{13}{116} a^{4} + \frac{2}{29} a^{2} - \frac{13}{29}$, $\frac{1}{9439616} a^{15} - \frac{11}{73747} a^{13} + \frac{548}{73747} a^{11} - \frac{47}{73747} a^{9} - \frac{9073}{589976} a^{7} + \frac{14787}{294988} a^{5} + \frac{33139}{147494} a^{3} - \frac{26772}{73747} a$, $\frac{1}{12571168310142108719287101629594756864} a^{16} + \frac{614416738283595399392912348403201}{6285584155071054359643550814797378432} a^{14} + \frac{753953031172900428846149533617401}{3142792077535527179821775407398689216} a^{12} + \frac{1065961826168705709951516461446755}{196424504845970448738860962962418076} a^{10} + \frac{3830914564121949143287454387451231}{196424504845970448738860962962418076} a^{8} + \frac{4008269164700992462605427079941399}{392849009691940897477721925924836152} a^{6} + \frac{17601849003949749123855169689955593}{196424504845970448738860962962418076} a^{4} + \frac{2518869068056518427957340607278721}{98212252422985224369430481481209038} a^{2} + \frac{4963927173635279906967660857424}{19310313099289269439526244884233}$, $\frac{1}{12571168310142108719287101629594756864} a^{17} - \frac{91958283098777763590791800935}{3142792077535527179821775407398689216} a^{15} - \frac{55748373266263420756054389804231}{3142792077535527179821775407398689216} a^{13} + \frac{22385174813118921069473534529014487}{1571396038767763589910887703699344608} a^{11} + \frac{18615067830618377912445618531279535}{785698019383881794955443851849672304} a^{9} - \frac{7215874049694637192642119211392481}{196424504845970448738860962962418076} a^{7} - \frac{9018832187908911096895918688449839}{98212252422985224369430481481209038} a^{5} + \frac{1228471445782847109532533393738780}{49106126211492612184715240740604519} a^{3} + \frac{7693144106446318495479026487227124}{49106126211492612184715240740604519} a$

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

Class group and class number

$C_{2}\times C_{791649756}$, which has order $1583299512$ (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:  \( 771793981.9189847 \) (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

\(\Q(\sqrt{-114}) \), 3.3.29241.2, 6.0.24953372987904.2, 9.9.532962204162830310969.5

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 R R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ $18$ $18$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
3Data not computed
19Data not computed