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Properties

Label 18.0.64776847223...6403.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 367^{8}$
Root discriminant $23.90$
Ramified primes $3, 367$
Class number $1$
Class group Trivial
Galois group $S_3\times S_4$ (as 18T69)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27, 0, 1539, 0, 1197, 0, -108, 0, 69, 0, 153, 0, 14, 0, -6, 0, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 3*x^16 - 6*x^14 + 14*x^12 + 153*x^10 + 69*x^8 - 108*x^6 + 1197*x^4 + 1539*x^2 + 27)
 
gp: K = bnfinit(x^18 + 3*x^16 - 6*x^14 + 14*x^12 + 153*x^10 + 69*x^8 - 108*x^6 + 1197*x^4 + 1539*x^2 + 27, 1)
 

Normalized defining polynomial

\( x^{18} + 3 x^{16} - 6 x^{14} + 14 x^{12} + 153 x^{10} + 69 x^{8} - 108 x^{6} + 1197 x^{4} + 1539 x^{2} + 27 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-6477684722397364709666403=-\,3^{9}\cdot 367^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.90$
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:  $3, 367$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{4}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{8} - \frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{18} a^{12} + \frac{1}{18} a^{10} + \frac{1}{18} a^{8} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{18} a^{13} + \frac{1}{18} a^{11} + \frac{1}{18} a^{9} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{558} a^{14} + \frac{5}{279} a^{12} + \frac{1}{18} a^{10} + \frac{29}{186} a^{8} - \frac{1}{6} a^{7} + \frac{1}{31} a^{6} + \frac{1}{6} a^{5} + \frac{19}{93} a^{4} - \frac{1}{2} a^{3} + \frac{21}{62} a^{2} + \frac{17}{62}$, $\frac{1}{558} a^{15} + \frac{5}{279} a^{13} + \frac{1}{18} a^{11} - \frac{1}{93} a^{9} + \frac{1}{31} a^{7} - \frac{1}{6} a^{6} + \frac{23}{62} a^{5} - \frac{1}{3} a^{4} + \frac{21}{62} a^{3} - \frac{1}{2} a^{2} - \frac{7}{31} a - \frac{1}{2}$, $\frac{1}{8016786} a^{16} - \frac{5}{296918} a^{14} + \frac{10997}{2672262} a^{12} + \frac{22348}{4008393} a^{10} - \frac{70486}{445377} a^{8} - \frac{51268}{445377} a^{6} - \frac{1}{2} a^{5} + \frac{111772}{445377} a^{4} - \frac{1}{2} a^{3} + \frac{212515}{890754} a^{2} - \frac{1}{2} a + \frac{61615}{148459}$, $\frac{1}{8016786} a^{17} - \frac{5}{296918} a^{15} + \frac{10997}{2672262} a^{13} + \frac{22348}{4008393} a^{11} + \frac{7487}{890754} a^{9} - \frac{1}{6} a^{8} - \frac{51268}{445377} a^{7} - \frac{1}{6} a^{6} + \frac{75085}{890754} a^{5} + \frac{1}{3} a^{4} + \frac{212515}{890754} a^{3} - \frac{25229}{296918} a - \frac{1}{2}$

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:  $8$
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:  \( 348966.988693 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times S_4$ (as 18T69):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 15 conjugacy class representatives for $S_3\times S_4$
Character table for $S_3\times S_4$

Intermediate fields

3.1.367.1, 3.3.1101.1, 6.0.3636603.2, 9.3.489810421467.1

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

Sibling fields

Degree 12 sibling: 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
367Data not computed