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Properties

Label 18.0.88654632917...8752.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{9}\cdot 107^{8}$
Root discriminant $27.64$
Ramified primes $2, 3, 107$
Class number $2$
Class group $[2]$
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, 81, 0, 144, 0, 117, 0, 60, 0, -63, 0, -44, 0, 11, 0, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 + 11*x^14 - 44*x^12 - 63*x^10 + 60*x^8 + 117*x^6 + 144*x^4 + 81*x^2 + 27)
 
gp: K = bnfinit(x^18 + 9*x^16 + 11*x^14 - 44*x^12 - 63*x^10 + 60*x^8 + 117*x^6 + 144*x^4 + 81*x^2 + 27, 1)
 

Normalized defining polynomial

\( x^{18} + 9 x^{16} + 11 x^{14} - 44 x^{12} - 63 x^{10} + 60 x^{8} + 117 x^{6} + 144 x^{4} + 81 x^{2} + 27 \)

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:  \(-88654632917746799944138752=-\,2^{18}\cdot 3^{9}\cdot 107^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.64$
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:  $2, 3, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{7} a^{8} - \frac{2}{7} a^{6} - \frac{2}{7} a^{4} + \frac{2}{7} a^{2} + \frac{1}{7}$, $\frac{1}{7} a^{9} - \frac{2}{7} a^{7} - \frac{2}{7} a^{5} + \frac{2}{7} a^{3} + \frac{1}{7} a$, $\frac{1}{21} a^{10} + \frac{8}{21} a^{6} - \frac{2}{21} a^{4} - \frac{3}{7} a^{2} + \frac{3}{7}$, $\frac{1}{21} a^{11} + \frac{8}{21} a^{7} - \frac{2}{21} a^{5} - \frac{3}{7} a^{3} + \frac{3}{7} a$, $\frac{1}{21} a^{12} - \frac{1}{21} a^{8} - \frac{5}{21} a^{6} + \frac{3}{7} a^{4} - \frac{3}{7} a^{2} - \frac{3}{7}$, $\frac{1}{21} a^{13} - \frac{1}{21} a^{9} - \frac{5}{21} a^{7} + \frac{3}{7} a^{5} - \frac{3}{7} a^{3} - \frac{3}{7} a$, $\frac{1}{63} a^{14} - \frac{1}{63} a^{10} + \frac{1}{63} a^{8} - \frac{8}{21} a^{6} + \frac{1}{3} a^{4} - \frac{2}{7} a^{2} + \frac{3}{7}$, $\frac{1}{63} a^{15} - \frac{1}{63} a^{11} + \frac{1}{63} a^{9} - \frac{8}{21} a^{7} + \frac{1}{3} a^{5} - \frac{2}{7} a^{3} + \frac{3}{7} a$, $\frac{1}{3087} a^{16} + \frac{1}{1029} a^{14} - \frac{1}{441} a^{12} - \frac{2}{3087} a^{10} - \frac{17}{1029} a^{8} + \frac{122}{1029} a^{6} + \frac{16}{49} a^{4} + \frac{30}{343} a^{2} - \frac{171}{343}$, $\frac{1}{3087} a^{17} + \frac{1}{1029} a^{15} - \frac{1}{441} a^{13} - \frac{2}{3087} a^{11} - \frac{17}{1029} a^{9} + \frac{122}{1029} a^{7} + \frac{16}{49} a^{5} + \frac{30}{343} a^{3} - \frac{171}{343} a$

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:  \( 189847.945068 \)
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

3.1.107.1, 3.3.321.1, 6.0.19783872.2, 9.3.3539149227.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.12.12.11$x^{12} - 6 x^{10} - 73 x^{8} + 140 x^{6} + 79 x^{4} - 6 x^{2} + 57$$2$$6$$12$$A_4 \times C_2$$[2, 2]^{6}$
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.12.9.2$x^{12} - 9 x^{4} + 27$$4$$3$$9$$D_4 \times C_3$$[\ ]_{4}^{6}$
107Data not computed