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Properties

Label 18.0.11437311792...1824.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{45}\cdot 19^{16}$
Root discriminant $603.97$
Ramified primes $2, 3, 19$
Class number $2864269422$ (GRH)
Class group $[7, 409181346]$ (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![1084361854375, -38029861350, 135663898545, 20624060568, 9735437304, 2758880142, 2434990194, 95193990, 4537944, 37696646, 13658103, -332424, -134160, 114912, 25407, -684, -288, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 288*x^16 - 684*x^15 + 25407*x^14 + 114912*x^13 - 134160*x^12 - 332424*x^11 + 13658103*x^10 + 37696646*x^9 + 4537944*x^8 + 95193990*x^7 + 2434990194*x^6 + 2758880142*x^5 + 9735437304*x^4 + 20624060568*x^3 + 135663898545*x^2 - 38029861350*x + 1084361854375)
 
gp: K = bnfinit(x^18 - 288*x^16 - 684*x^15 + 25407*x^14 + 114912*x^13 - 134160*x^12 - 332424*x^11 + 13658103*x^10 + 37696646*x^9 + 4537944*x^8 + 95193990*x^7 + 2434990194*x^6 + 2758880142*x^5 + 9735437304*x^4 + 20624060568*x^3 + 135663898545*x^2 - 38029861350*x + 1084361854375, 1)
 

Normalized defining polynomial

\( x^{18} - 288 x^{16} - 684 x^{15} + 25407 x^{14} + 114912 x^{13} - 134160 x^{12} - 332424 x^{11} + 13658103 x^{10} + 37696646 x^{9} + 4537944 x^{8} + 95193990 x^{7} + 2434990194 x^{6} + 2758880142 x^{5} + 9735437304 x^{4} + 20624060568 x^{3} + 135663898545 x^{2} - 38029861350 x + 1084361854375 \)

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:  \(-114373117921862173420296089417336295857933279821824=-\,2^{27}\cdot 3^{45}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $603.97$
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$$\chi_{4104}(1,·)$, $\chi_{4104}(577,·)$, $\chi_{4104}(3341,·)$, $\chi_{4104}(3725,·)$, $\chi_{4104}(3217,·)$, $\chi_{4104}(1493,·)$, $\chi_{4104}(2905,·)$, $\chi_{4104}(1753,·)$, $\chi_{4104}(461,·)$, $\chi_{4104}(3749,·)$, $\chi_{4104}(2981,·)$, $\chi_{4104}(3505,·)$, $\chi_{4104}(1897,·)$, $\chi_{4104}(365,·)$, $\chi_{4104}(1201,·)$, $\chi_{4104}(2933,·)$, $\chi_{4104}(505,·)$, $\chi_{4104}(1301,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{4}$, $\frac{1}{45} a^{9} + \frac{1}{3} a^{3} + \frac{1}{5} a + \frac{2}{9}$, $\frac{1}{225} a^{10} + \frac{2}{25} a^{6} - \frac{1}{3} a^{4} + \frac{9}{25} a^{2} + \frac{4}{9} a$, $\frac{1}{675} a^{11} - \frac{1}{675} a^{10} - \frac{1}{135} a^{9} + \frac{1}{15} a^{8} + \frac{7}{75} a^{7} - \frac{2}{75} a^{6} + \frac{1}{45} a^{5} + \frac{17}{45} a^{4} - \frac{13}{225} a^{3} + \frac{19}{675} a^{2} - \frac{47}{135} a - \frac{2}{27}$, $\frac{1}{675} a^{12} - \frac{1}{135} a^{9} - \frac{1}{25} a^{8} + \frac{1}{15} a^{7} - \frac{2}{45} a^{6} - \frac{11}{75} a^{4} - \frac{4}{135} a^{3} - \frac{2}{5} a^{2} + \frac{4}{15} a + \frac{7}{27}$, $\frac{1}{3375} a^{13} - \frac{2}{3375} a^{12} + \frac{2}{3375} a^{11} + \frac{1}{675} a^{10} + \frac{11}{1125} a^{9} - \frac{8}{125} a^{8} + \frac{47}{1125} a^{7} + \frac{16}{225} a^{6} + \frac{22}{1125} a^{5} + \frac{868}{3375} a^{4} + \frac{457}{3375} a^{3} + \frac{76}{675} a^{2} + \frac{38}{135} a + \frac{4}{9}$, $\frac{1}{16875} a^{14} - \frac{4}{5625} a^{12} - \frac{11}{16875} a^{11} + \frac{2}{1875} a^{10} - \frac{1}{225} a^{9} + \frac{368}{5625} a^{8} + \frac{6}{625} a^{7} - \frac{181}{1875} a^{6} + \frac{11}{135} a^{5} + \frac{104}{625} a^{4} - \frac{1192}{5625} a^{3} - \frac{868}{3375} a^{2} + \frac{17}{75} a - \frac{1}{9}$, $\frac{1}{421875} a^{15} - \frac{1}{46875} a^{14} - \frac{52}{421875} a^{13} + \frac{302}{421875} a^{12} - \frac{88}{421875} a^{11} - \frac{29}{140625} a^{10} + \frac{142}{15625} a^{9} - \frac{1}{46875} a^{8} + \frac{5341}{140625} a^{7} - \frac{38264}{421875} a^{6} + \frac{13306}{140625} a^{5} - \frac{101068}{421875} a^{4} - \frac{82561}{421875} a^{3} - \frac{29093}{84375} a^{2} + \frac{811}{5625} a - \frac{53}{225}$, $\frac{1}{19782106241765625} a^{16} - \frac{11252597213}{19782106241765625} a^{15} - \frac{245045380241}{19782106241765625} a^{14} + \frac{472412720932}{3956421248353125} a^{13} - \frac{8703545347396}{19782106241765625} a^{12} + \frac{2763527908228}{3956421248353125} a^{11} + \frac{5151986080669}{6594035413921875} a^{10} + \frac{37476274545196}{3956421248353125} a^{9} - \frac{596096005524247}{6594035413921875} a^{8} + \frac{184460109313369}{19782106241765625} a^{7} - \frac{1669781795538626}{19782106241765625} a^{6} + \frac{45263415596132}{3956421248353125} a^{5} + \frac{202382386683271}{482490396140625} a^{4} + \frac{4098409547601629}{19782106241765625} a^{3} + \frac{276579192437962}{3956421248353125} a^{2} + \frac{63979347927581}{263761416556875} a + \frac{4196676620786}{31651369986825}$, $\frac{1}{104766617680581639762906033903050913491296973738626171875} a^{17} + \frac{364138789414390302141676046770541802808}{34922205893527213254302011301016971163765657912875390625} a^{16} + \frac{69221422187328896384142875978339396965597722060538}{104766617680581639762906033903050913491296973738626171875} a^{15} - \frac{587003151687005800554624533526211914182624141617097}{104766617680581639762906033903050913491296973738626171875} a^{14} - \frac{901907972422986731008154737557663242649010896577507}{34922205893527213254302011301016971163765657912875390625} a^{13} + \frac{77585070872036141018727280743064968738363699364522883}{104766617680581639762906033903050913491296973738626171875} a^{12} - \frac{77560260478019724480312033243798534086126463386923318}{104766617680581639762906033903050913491296973738626171875} a^{11} + \frac{185818096649377326861282337438523755812628425666913744}{104766617680581639762906033903050913491296973738626171875} a^{10} - \frac{97164206053642185880293496333955115249223543098983947}{34922205893527213254302011301016971163765657912875390625} a^{9} - \frac{8218331012025985680363108043787990788247861471457892313}{104766617680581639762906033903050913491296973738626171875} a^{8} - \frac{1516341565752189197067385637817059896456239465474841106}{34922205893527213254302011301016971163765657912875390625} a^{7} + \frac{8163608825688492718124388218520411192780123903224523358}{104766617680581639762906033903050913491296973738626171875} a^{6} + \frac{2028225886647553191043246251409348942233689278335395311}{104766617680581639762906033903050913491296973738626171875} a^{5} - \frac{8106066331652452069301115752362371676501208932650577748}{34922205893527213254302011301016971163765657912875390625} a^{4} - \frac{15046391513485913016061154724302852629473235674222533227}{104766617680581639762906033903050913491296973738626171875} a^{3} + \frac{2540856089152034738985736312338938259475194947560175779}{20953323536116327952581206780610182698259394747725234375} a^{2} - \frac{33805931113015488260655527009074158282380658856597304}{4190664707223265590516241356122036539651878949545046875} a - \frac{1839825057134036004784565215351925583411028443416329}{6208392158849282356320357564625239318002783628955625}$

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

Class group and class number

$C_{7}\times C_{409181346}$, which has order $2864269422$ (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:  \( 5772307958.489205 \) (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

\(\Q(\sqrt{-6}) \), 3.3.29241.2, 6.0.1313335420416.1, 9.9.532962204162830310969.6

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 ${\href{/LocalNumberField/5.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ $18$ $18$ R $18$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ $18$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
3Data not computed
19Data not computed