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Properties

Label 18.0.21730892405...4656.6
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 $29703669048$ (GRH)
Class group $[2, 14851834524]$ (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![40292160000, 0, 755493868800, 0, 363270583296, 0, 70338466944, 0, 6981100992, 0, 377384688, 0, 10824528, 0, 145692, 0, 684, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 684*x^16 + 145692*x^14 + 10824528*x^12 + 377384688*x^10 + 6981100992*x^8 + 70338466944*x^6 + 363270583296*x^4 + 755493868800*x^2 + 40292160000)
 
gp: K = bnfinit(x^18 + 684*x^16 + 145692*x^14 + 10824528*x^12 + 377384688*x^10 + 6981100992*x^8 + 70338466944*x^6 + 363270583296*x^4 + 755493868800*x^2 + 40292160000, 1)
 

Normalized defining polynomial

\( x^{18} + 684 x^{16} + 145692 x^{14} + 10824528 x^{12} + 377384688 x^{10} + 6981100992 x^{8} + 70338466944 x^{6} + 363270583296 x^{4} + 755493868800 x^{2} + 40292160000 \)

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$$\chi_{4104}(1,·)$, $\chi_{4104}(2051,·)$, $\chi_{4104}(577,·)$, $\chi_{4104}(1547,·)$, $\chi_{4104}(3217,·)$, $\chi_{4104}(1475,·)$, $\chi_{4104}(2905,·)$, $\chi_{4104}(1753,·)$, $\chi_{4104}(851,·)$, $\chi_{4104}(2651,·)$, $\chi_{4104}(155,·)$, $\chi_{4104}(3505,·)$, $\chi_{4104}(1897,·)$, $\chi_{4104}(299,·)$, $\chi_{4104}(1201,·)$, $\chi_{4104}(3251,·)$, $\chi_{4104}(505,·)$, $\chi_{4104}(2939,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{20} a^{5} + \frac{1}{5} a$, $\frac{1}{40} a^{6} + \frac{1}{10} a^{2}$, $\frac{1}{40} a^{7} + \frac{1}{10} a^{3}$, $\frac{1}{80} a^{8} + \frac{1}{20} a^{4}$, $\frac{1}{80} a^{9} - \frac{1}{5} a$, $\frac{1}{800} a^{10} + \frac{1}{100} a^{6} + \frac{1}{50} a^{2}$, $\frac{1}{800} a^{11} + \frac{1}{100} a^{7} + \frac{1}{50} a^{3}$, $\frac{1}{8000} a^{12} - \frac{1}{2000} a^{10} - \frac{1}{250} a^{8} - \frac{9}{1000} a^{6} - \frac{17}{250} a^{4} - \frac{7}{250} a^{2} - \frac{1}{5}$, $\frac{1}{8000} a^{13} - \frac{1}{2000} a^{11} - \frac{1}{250} a^{9} - \frac{9}{1000} a^{7} - \frac{9}{500} a^{5} - \frac{7}{250} a^{3}$, $\frac{1}{3280000} a^{14} - \frac{3}{164000} a^{12} + \frac{483}{820000} a^{10} + \frac{473}{82000} a^{8} + \frac{769}{102500} a^{6} - \frac{281}{20500} a^{4} + \frac{681}{51250} a^{2} + \frac{368}{1025}$, $\frac{1}{770800000} a^{15} - \frac{1129}{19270000} a^{13} - \frac{559}{1025000} a^{11} + \frac{6833}{1204375} a^{9} - \frac{317237}{48175000} a^{7} + \frac{23146}{1204375} a^{5} - \frac{622519}{12043750} a^{3} - \frac{57032}{240875} a$, $\frac{1}{135536641794658793395693600000} a^{16} + \frac{1314745156783057746019}{13553664179465879339569360000} a^{14} + \frac{7381075833486005499489}{720939584014142518062200000} a^{12} + \frac{932872757277725058371103}{3388416044866469834892340000} a^{10} - \frac{1506234100873151040161107}{529440007010385911701928125} a^{8} + \frac{2453926216385606261339301}{211776002804154364680771250} a^{6} - \frac{125012785828090592635434069}{2117760028041543646807712500} a^{4} - \frac{1378828301124647611501303}{21177600280415436468077125} a^{2} + \frac{8772042477611589612439}{18023489600353562951555}$, $\frac{1}{135536641794658793395693600000} a^{17} - \frac{2023382597016113023}{6776832089732939669784680000} a^{15} + \frac{62207287029952801754663}{2117760028041543646807712500} a^{13} - \frac{122404214523908143341471}{1694208022433234917446170000} a^{11} - \frac{7252178810713092249447337}{8471040112166174587230850000} a^{9} + \frac{99443038349174964545921}{211776002804154364680771250} a^{7} - \frac{43203726934778444701668569}{2117760028041543646807712500} a^{5} - \frac{1027062535802625146285832}{21177600280415436468077125} a^{3} + \frac{37661171144786020386702}{847104011216617458723085} a$

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

Class group and class number

$C_{2}\times C_{14851834524}$, which has order $29703669048$ (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

\(\Q(\sqrt{-114}) \), 3.3.29241.2, 6.0.24953372987904.2, 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 ${\href{/LocalNumberField/5.1.0.1}{1} }^{18}$ $18$ $18$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{18}$ $18$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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