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Properties

Label 18.0.72436308017...1552.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{44}\cdot 19^{17}$
Root discriminant $669.20$
Ramified primes $2, 3, 19$
Class number $2355691086$ (GRH)
Class group $[9, 261743454]$ (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![985095890432, 0, 3069202731264, 0, 1120116736512, 0, 168726690432, 0, 13065198912, 0, 553659696, 0, 12701424, 0, 145692, 0, 684, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 684*x^16 + 145692*x^14 + 12701424*x^12 + 553659696*x^10 + 13065198912*x^8 + 168726690432*x^6 + 1120116736512*x^4 + 3069202731264*x^2 + 985095890432)
 
gp: K = bnfinit(x^18 + 684*x^16 + 145692*x^14 + 12701424*x^12 + 553659696*x^10 + 13065198912*x^8 + 168726690432*x^6 + 1120116736512*x^4 + 3069202731264*x^2 + 985095890432, 1)
 

Normalized defining polynomial

\( x^{18} + 684 x^{16} + 145692 x^{14} + 12701424 x^{12} + 553659696 x^{10} + 13065198912 x^{8} + 168726690432 x^{6} + 1120116736512 x^{4} + 3069202731264 x^{2} + 985095890432 \)

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:  \(-724363080171793764995208566309796540433577438871552=-\,2^{27}\cdot 3^{44}\cdot 19^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $669.20$
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$$\chi_{4104}(1537,·)$, $\chi_{4104}(1093,·)$, $\chi_{4104}(1,·)$, $\chi_{4104}(2569,·)$, $\chi_{4104}(781,·)$, $\chi_{4104}(529,·)$, $\chi_{4104}(2197,·)$, $\chi_{4104}(421,·)$, $\chi_{4104}(481,·)$, $\chi_{4104}(3301,·)$, $\chi_{4104}(2029,·)$, $\chi_{4104}(1405,·)$, $\chi_{4104}(385,·)$, $\chi_{4104}(3637,·)$, $\chi_{4104}(577,·)$, $\chi_{4104}(505,·)$, $\chi_{4104}(769,·)$, $\chi_{4104}(2749,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{22208} a^{13} + \frac{81}{11104} a^{11} - \frac{25}{1388} a^{9} + \frac{2}{347} a^{7} + \frac{67}{694} a^{5} - \frac{37}{694} a^{3} + \frac{110}{347} a$, $\frac{1}{44416} a^{14} + \frac{81}{22208} a^{12} - \frac{25}{2776} a^{10} + \frac{1}{347} a^{8} + \frac{67}{1388} a^{6} - \frac{37}{1388} a^{4} + \frac{55}{347} a^{2}$, $\frac{1}{1288064} a^{15} + \frac{5}{644032} a^{13} - \frac{523}{80504} a^{11} + \frac{131}{5552} a^{9} + \frac{4817}{80504} a^{7} + \frac{1309}{20126} a^{5} - \frac{1589}{20126} a^{3} + \frac{2744}{10063} a$, $\frac{1}{6796903430452939337812168070542144256} a^{16} - \frac{35612984788685978384844241665897}{3398451715226469668906084035271072128} a^{14} + \frac{355970719479810317824901207185137}{1699225857613234834453042017635536064} a^{12} - \frac{416803752931959974511709005273289}{29296997545055773007811069269578208} a^{10} - \frac{781676459888564007252952395210011}{212403232201654354306630252204442008} a^{8} - \frac{3783382203065246615440303787115615}{106201616100827177153315126102221004} a^{6} + \frac{3960384063928612215679199552023151}{53100808050413588576657563051110502} a^{4} - \frac{578558748282031873491562498377533}{26550404025206794288328781525555251} a^{2} + \frac{1094921716329788684668158121}{7603511143544028324245661991}$, $\frac{1}{6796903430452939337812168070542144256} a^{17} - \frac{82096626259929163385753776531}{212403232201654354306630252204442008} a^{15} - \frac{8131015139939022509926562915889}{1699225857613234834453042017635536064} a^{13} + \frac{5526772181795237522314860536889471}{849612928806617417226521008817768032} a^{11} - \frac{222045023021981281075305706401330}{26550404025206794288328781525555251} a^{9} - \frac{10218374864774319948536418508662615}{212403232201654354306630252204442008} a^{7} - \frac{603961615305167732567894556797285}{106201616100827177153315126102221004} a^{5} - \frac{9291361321438608792289458440388857}{53100808050413588576657563051110502} a^{3} + \frac{35950110271107532409017200840112}{76514132637483557026884096615433} a$

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

Class group and class number

$C_{9}\times C_{261743454}$, which has order $2355691086$ (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

\(\Q(\sqrt{-38}) \), 3.3.29241.2, 6.0.8317790995968.4, 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{18}$ $18$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$3$3.9.22.4$x^{9} + 9 x^{8} + 9 x^{7} + 12 x^{6} + 18 x^{5} + 33$$9$$1$$22$$C_9$$[2, 3]$
3.9.22.4$x^{9} + 9 x^{8} + 9 x^{7} + 12 x^{6} + 18 x^{5} + 33$$9$$1$$22$$C_9$$[2, 3]$
19Data not computed