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Properties

Label 18.6.10419354122...4352.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{26}\cdot 11^{4}\cdot 13^{9}$
Root discriminant $16.72$
Ramified primes $2, 11, 13$
Class number $1$
Class group Trivial
Galois group $C_3\wr S_3:C_2$ (as 18T137)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -6, -6, -2, 16, 9, -20, -50, -92, -50, -20, 9, 16, -2, -6, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^16 - 6*x^15 - 2*x^14 + 16*x^13 + 9*x^12 - 20*x^11 - 50*x^10 - 92*x^9 - 50*x^8 - 20*x^7 + 9*x^6 + 16*x^5 - 2*x^4 - 6*x^3 - 6*x^2 + 1)
 
gp: K = bnfinit(x^18 - 6*x^16 - 6*x^15 - 2*x^14 + 16*x^13 + 9*x^12 - 20*x^11 - 50*x^10 - 92*x^9 - 50*x^8 - 20*x^7 + 9*x^6 + 16*x^5 - 2*x^4 - 6*x^3 - 6*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{16} - 6 x^{15} - 2 x^{14} + 16 x^{13} + 9 x^{12} - 20 x^{11} - 50 x^{10} - 92 x^{9} - 50 x^{8} - 20 x^{7} + 9 x^{6} + 16 x^{5} - 2 x^{4} - 6 x^{3} - 6 x^{2} + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10419354122831477604352=2^{26}\cdot 11^{4}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.72$
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:  $2, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{2} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{2109512} a^{16} - \frac{30420}{263689} a^{15} + \frac{67571}{2109512} a^{14} + \frac{432795}{1054756} a^{13} + \frac{17189}{2109512} a^{12} + \frac{118219}{1054756} a^{11} + \frac{3915}{1054756} a^{10} - \frac{427961}{1054756} a^{9} - \frac{32983}{263689} a^{8} - \frac{427961}{1054756} a^{7} - \frac{523463}{1054756} a^{6} + \frac{118219}{1054756} a^{5} + \frac{544567}{2109512} a^{4} - \frac{94583}{1054756} a^{3} - \frac{459807}{2109512} a^{2} + \frac{202849}{527378} a + \frac{527379}{2109512}$, $\frac{1}{4219024} a^{17} - \frac{1}{4219024} a^{16} - \frac{284025}{4219024} a^{15} + \frac{175783}{4219024} a^{14} - \frac{405785}{4219024} a^{13} - \frac{1937519}{4219024} a^{12} - \frac{195249}{1054756} a^{11} + \frac{202617}{1054756} a^{10} + \frac{951399}{2109512} a^{9} + \frac{32469}{2109512} a^{8} - \frac{61861}{263689} a^{7} + \frac{106009}{527378} a^{6} - \frac{244259}{4219024} a^{5} - \frac{1835745}{4219024} a^{4} + \frac{171975}{4219024} a^{3} - \frac{405789}{4219024} a^{2} - \frac{878973}{4219024} a + \frac{770737}{4219024}$

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:  $11$
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:  \( 14879.7693607 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\wr S_3:C_2$ (as 18T137):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 324
The 17 conjugacy class representatives for $C_3\wr S_3:C_2$
Character table for $C_3\wr S_3:C_2$

Intermediate fields

\(\Q(\sqrt{13}) \), 3.1.104.1, 6.2.140608.1, 9.3.28310577152.1 x3

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

Sibling fields

Degree 9 siblings: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.12.22.79$x^{12} + 2 x^{10} + 4 x^{8} + 4 x^{6} + 4 x^{4} + 4$$6$$2$$22$$D_6$$[3]_{3}^{2}$
$11$11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$