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Properties

Label 18.18.2766879715...3728.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{18}\cdot 3^{27}\cdot 7^{12}$
Root discriminant $38.03$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-27, 0, 486, 0, -2619, 0, 6264, 0, -7668, 0, 5040, 0, -1788, 0, 333, 0, -30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 30*x^16 + 333*x^14 - 1788*x^12 + 5040*x^10 - 7668*x^8 + 6264*x^6 - 2619*x^4 + 486*x^2 - 27)
 
gp: K = bnfinit(x^18 - 30*x^16 + 333*x^14 - 1788*x^12 + 5040*x^10 - 7668*x^8 + 6264*x^6 - 2619*x^4 + 486*x^2 - 27, 1)
 

Normalized defining polynomial

\( x^{18} - 30 x^{16} + 333 x^{14} - 1788 x^{12} + 5040 x^{10} - 7668 x^{8} + 6264 x^{6} - 2619 x^{4} + 486 x^{2} - 27 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(27668797159880354103659593728=2^{18}\cdot 3^{27}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.03$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(252=2^{2}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{252}(1,·)$, $\chi_{252}(193,·)$, $\chi_{252}(11,·)$, $\chi_{252}(205,·)$, $\chi_{252}(85,·)$, $\chi_{252}(23,·)$, $\chi_{252}(25,·)$, $\chi_{252}(71,·)$, $\chi_{252}(155,·)$, $\chi_{252}(95,·)$, $\chi_{252}(37,·)$, $\chi_{252}(169,·)$, $\chi_{252}(107,·)$, $\chi_{252}(109,·)$, $\chi_{252}(239,·)$, $\chi_{252}(179,·)$, $\chi_{252}(121,·)$, $\chi_{252}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{3} a^{11}$, $\frac{1}{9} a^{12}$, $\frac{1}{9} a^{13}$, $\frac{1}{36} a^{14} + \frac{1}{4}$, $\frac{1}{36} a^{15} + \frac{1}{4} a$, $\frac{1}{44208} a^{16} - \frac{47}{4912} a^{14} - \frac{109}{11052} a^{12} - \frac{101}{614} a^{10} + \frac{173}{1842} a^{8} - \frac{103}{1228} a^{6} + \frac{129}{1228} a^{4} - \frac{1687}{4912} a^{2} - \frac{75}{4912}$, $\frac{1}{44208} a^{17} - \frac{47}{4912} a^{15} - \frac{109}{11052} a^{13} - \frac{101}{614} a^{11} + \frac{173}{1842} a^{9} - \frac{103}{1228} a^{7} + \frac{129}{1228} a^{5} - \frac{1687}{4912} a^{3} - \frac{75}{4912} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  $17$
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:  \( 278806740.442 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\zeta_{7})^+\), 3.3.3969.2, \(\Q(\zeta_{9})^+\), 3.3.3969.1, 6.6.4148928.1, 6.6.3024568512.2, \(\Q(\zeta_{36})^+\), 6.6.3024568512.1, 9.9.62523502209.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
7Data not computed