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Properties

Label 18.0.65503383620...0304.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{33}\cdot 3^{27}$
Root discriminant $18.52$
Ramified primes $2, 3$
Class number $2$
Class group $[2]$
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![216, 0, 324, 0, -324, 0, -315, 0, 54, 0, 27, 0, 48, 0, 27, 0, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 6*x^16 + 27*x^14 + 48*x^12 + 27*x^10 + 54*x^8 - 315*x^6 - 324*x^4 + 324*x^2 + 216)
 
gp: K = bnfinit(x^18 + 6*x^16 + 27*x^14 + 48*x^12 + 27*x^10 + 54*x^8 - 315*x^6 - 324*x^4 + 324*x^2 + 216, 1)
 

Normalized defining polynomial

\( x^{18} + 6 x^{16} + 27 x^{14} + 48 x^{12} + 27 x^{10} + 54 x^{8} - 315 x^{6} - 324 x^{4} + 324 x^{2} + 216 \)

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:  \(-65503383620958031970304=-\,2^{33}\cdot 3^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.52$
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:  $2, 3$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{10} - \frac{1}{12} a^{6}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{7}$, $\frac{1}{72} a^{12} - \frac{1}{24} a^{10} - \frac{1}{24} a^{8} + \frac{1}{24} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{144} a^{13} - \frac{1}{144} a^{12} - \frac{1}{48} a^{11} + \frac{1}{48} a^{10} + \frac{1}{48} a^{9} - \frac{1}{48} a^{8} - \frac{1}{16} a^{7} + \frac{1}{16} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{144} a^{14} - \frac{1}{24} a^{10} + \frac{1}{16} a^{6} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{288} a^{15} - \frac{1}{288} a^{14} + \frac{1}{48} a^{11} - \frac{1}{48} a^{10} - \frac{1}{24} a^{9} - \frac{1}{24} a^{8} - \frac{1}{96} a^{7} + \frac{1}{96} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{170784} a^{16} - \frac{401}{170784} a^{14} - \frac{217}{85392} a^{12} + \frac{193}{28464} a^{10} - \frac{541}{56928} a^{8} + \frac{1981}{56928} a^{6} - \frac{73}{593} a^{4} - \frac{707}{4744} a^{2} + \frac{373}{2372}$, $\frac{1}{341568} a^{17} - \frac{1}{341568} a^{16} - \frac{401}{341568} a^{15} + \frac{401}{341568} a^{14} - \frac{217}{170784} a^{13} + \frac{217}{170784} a^{12} + \frac{193}{56928} a^{11} - \frac{193}{56928} a^{10} + \frac{1401}{37952} a^{9} - \frac{1401}{37952} a^{8} - \frac{7507}{113856} a^{7} + \frac{7507}{113856} a^{6} - \frac{885}{4744} a^{5} + \frac{885}{4744} a^{4} - \frac{707}{9488} a^{3} + \frac{707}{9488} a^{2} - \frac{1999}{4744} a + \frac{1999}{4744}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3117.4002199827046 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-6}) \), \(\Q(\zeta_{9})^+\), 3.1.648.1, 6.0.40310784.2, 6.0.10077696.1, 9.3.272097792.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 sibling: data not computed

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.12.24.307$x^{12} + 28 x^{11} - 2 x^{10} + 16 x^{9} + 26 x^{8} + 8 x^{7} + 20 x^{6} - 24 x^{5} - 8 x^{4} + 32 x^{3} + 32 x^{2} + 32 x + 24$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
3Data not computed