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Properties

Label 18.2.49072127903...8528.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{47}\cdot 3^{20}$
Root discriminant $20.71$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $S_3\wr C_2$ (as 18T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 0, 33, 0, -180, 0, -72, 0, 24, 0, -18, 0, -60, 0, 24, 0, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^16 + 24*x^14 - 60*x^12 - 18*x^10 + 24*x^8 - 72*x^6 - 180*x^4 + 33*x^2 - 2)
 
gp: K = bnfinit(x^18 - 6*x^16 + 24*x^14 - 60*x^12 - 18*x^10 + 24*x^8 - 72*x^6 - 180*x^4 + 33*x^2 - 2, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{16} + 24 x^{14} - 60 x^{12} - 18 x^{10} + 24 x^{8} - 72 x^{6} - 180 x^{4} + 33 x^{2} - 2 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(490721279033276815638528=2^{47}\cdot 3^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{16} a^{10} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{5}{16} a^{2} - \frac{1}{8}$, $\frac{1}{16} a^{11} - \frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a^{4} + \frac{5}{16} a^{3} - \frac{1}{2} a^{2} - \frac{1}{8} a$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{5}{16} a^{4} - \frac{1}{2} a^{3} - \frac{1}{8} a^{2}$, $\frac{1}{16} a^{13} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} + \frac{3}{16} a^{5} - \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{128} a^{14} - \frac{1}{32} a^{12} - \frac{1}{128} a^{10} + \frac{3}{64} a^{8} + \frac{27}{128} a^{6} - \frac{1}{2} a^{5} - \frac{3}{16} a^{4} - \frac{1}{2} a^{3} + \frac{29}{128} a^{2} - \frac{1}{64}$, $\frac{1}{256} a^{15} - \frac{1}{256} a^{14} + \frac{1}{64} a^{13} - \frac{1}{64} a^{12} - \frac{1}{256} a^{11} + \frac{1}{256} a^{10} - \frac{5}{128} a^{9} + \frac{5}{128} a^{8} + \frac{11}{256} a^{7} - \frac{11}{256} a^{6} - \frac{7}{16} a^{5} + \frac{7}{16} a^{4} - \frac{115}{256} a^{3} + \frac{115}{256} a^{2} - \frac{1}{128} a + \frac{1}{128}$, $\frac{1}{48896} a^{16} + \frac{15}{48896} a^{14} - \frac{1189}{48896} a^{12} - \frac{581}{48896} a^{10} + \frac{1533}{48896} a^{8} + \frac{10825}{48896} a^{6} + \frac{2637}{48896} a^{4} - \frac{12035}{48896} a^{2} - \frac{1819}{24448}$, $\frac{1}{97792} a^{17} - \frac{1}{97792} a^{16} + \frac{15}{97792} a^{15} - \frac{15}{97792} a^{14} + \frac{1867}{97792} a^{13} - \frac{1867}{97792} a^{12} + \frac{2475}{97792} a^{11} - \frac{2475}{97792} a^{10} - \frac{4579}{97792} a^{9} + \frac{4579}{97792} a^{8} + \frac{23049}{97792} a^{7} - \frac{23049}{97792} a^{6} + \frac{11805}{97792} a^{5} - \frac{11805}{97792} a^{4} + \frac{21581}{97792} a^{3} - \frac{21581}{97792} a^{2} - \frac{4875}{48896} a + \frac{4875}{48896}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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:  \( 279130.496866 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\wr C_2$ (as 18T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $S_3\wr C_2$
Character table for $S_3\wr C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), 9.1.30958682112.1

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

Sibling fields

Degree 6 siblings: 6.2.5971968.3, 6.2.1492992.1
Degree 9 sibling: 9.1.30958682112.1
Degree 12 siblings: 12.4.570630428688384.1, 12.0.1711891286065152.5, 12.4.570630428688384.2, 12.0.6847565144260608.26, 12.0.213986410758144.1, 12.0.3423782572130304.4
Degree 18 siblings: Deg 18, Deg 18

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.8.22.83$x^{8} + 4 x^{7} + 2 x^{4} + 4 x^{2} + 14$$8$$1$$22$$D_4$$[2, 3, 7/2]$
2.8.22.83$x^{8} + 4 x^{7} + 2 x^{4} + 4 x^{2} + 14$$8$$1$$22$$D_4$$[2, 3, 7/2]$
3Data not computed