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Properties

Label 18.0.12059532840...9408.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{9}\cdot 37^{17}$
Root discriminant $148.31$
Ramified primes $2, 3, 37$
Class number $2804004$ (GRH)
Class group $[2, 1402002]$ (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![372874752, 0, 2672269056, 0, 2599765632, 0, 964986048, 0, 172627200, 0, 15920064, 0, 759240, 0, 18648, 0, 222, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 222*x^16 + 18648*x^14 + 759240*x^12 + 15920064*x^10 + 172627200*x^8 + 964986048*x^6 + 2599765632*x^4 + 2672269056*x^2 + 372874752)
 
gp: K = bnfinit(x^18 + 222*x^16 + 18648*x^14 + 759240*x^12 + 15920064*x^10 + 172627200*x^8 + 964986048*x^6 + 2599765632*x^4 + 2672269056*x^2 + 372874752, 1)
 

Normalized defining polynomial

\( x^{18} + 222 x^{16} + 18648 x^{14} + 759240 x^{12} + 15920064 x^{10} + 172627200 x^{8} + 964986048 x^{6} + 2599765632 x^{4} + 2672269056 x^{2} + 372874752 \)

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:  \(-1205953284095546800596778947408883089408=-\,2^{27}\cdot 3^{9}\cdot 37^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $148.31$
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:  $2, 3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(888=2^{3}\cdot 3\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{888}(1,·)$, $\chi_{888}(49,·)$, $\chi_{888}(77,·)$, $\chi_{888}(145,·)$, $\chi_{888}(793,·)$, $\chi_{888}(601,·)$, $\chi_{888}(101,·)$, $\chi_{888}(221,·)$, $\chi_{888}(485,·)$, $\chi_{888}(673,·)$, $\chi_{888}(677,·)$, $\chi_{888}(625,·)$, $\chi_{888}(173,·)$, $\chi_{888}(317,·)$, $\chi_{888}(433,·)$, $\chi_{888}(437,·)$, $\chi_{888}(121,·)$, $\chi_{888}(509,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{6} a^{2}$, $\frac{1}{6} a^{3}$, $\frac{1}{36} a^{4}$, $\frac{1}{36} a^{5}$, $\frac{1}{216} a^{6}$, $\frac{1}{216} a^{7}$, $\frac{1}{1296} a^{8}$, $\frac{1}{1296} a^{9}$, $\frac{1}{7776} a^{10}$, $\frac{1}{7776} a^{11}$, $\frac{1}{46656} a^{12}$, $\frac{1}{46656} a^{13}$, $\frac{1}{12037248} a^{14} + \frac{7}{167184} a^{10} - \frac{13}{55728} a^{8} - \frac{5}{4644} a^{6} + \frac{11}{1548} a^{4} - \frac{11}{258} a^{2} - \frac{21}{43}$, $\frac{1}{12037248} a^{15} + \frac{7}{167184} a^{11} - \frac{13}{55728} a^{9} - \frac{5}{4644} a^{7} + \frac{11}{1548} a^{5} - \frac{11}{258} a^{3} - \frac{21}{43} a$, $\frac{1}{4651388858416896} a^{16} + \frac{31484785}{775231476402816} a^{14} - \frac{1186318945}{129205246067136} a^{12} + \frac{1577935}{1196344870992} a^{10} + \frac{454901551}{3589034612976} a^{8} + \frac{184057103}{199390811832} a^{6} - \frac{348647267}{33231801972} a^{4} - \frac{426340094}{8307950493} a^{2} - \frac{7266499}{2769316831}$, $\frac{1}{4651388858416896} a^{17} + \frac{31484785}{775231476402816} a^{15} - \frac{1186318945}{129205246067136} a^{13} + \frac{1577935}{1196344870992} a^{11} + \frac{454901551}{3589034612976} a^{9} + \frac{184057103}{199390811832} a^{7} - \frac{348647267}{33231801972} a^{5} - \frac{426340094}{8307950493} a^{3} - \frac{7266499}{2769316831} a$

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

Class group and class number

$C_{2}\times C_{1402002}$, which has order $2804004$ (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:  \( 409151.3102125697 \) (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

\(\Q(\sqrt{-222}) \), 3.3.1369.1, 6.0.958610861568.3, 9.9.3512479453921.1

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 $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ R $18$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
37Data not computed