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Properties

Label 18.0.29706609978...2208.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 19^{12}$
Root discriminant $20.14$
Ramified primes $2, 19$
Class number $1$
Class group Trivial
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, 0, 72, 0, 218, 0, 369, 0, 500, 0, 362, 0, 176, 0, 45, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^16 + 45*x^14 + 176*x^12 + 362*x^10 + 500*x^8 + 369*x^6 + 218*x^4 + 72*x^2 + 8)
 
gp: K = bnfinit(x^18 - 4*x^16 + 45*x^14 + 176*x^12 + 362*x^10 + 500*x^8 + 369*x^6 + 218*x^4 + 72*x^2 + 8, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{16} + 45 x^{14} + 176 x^{12} + 362 x^{10} + 500 x^{8} + 369 x^{6} + 218 x^{4} + 72 x^{2} + 8 \)

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:  \(-297066099785564011102208=-\,2^{27}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.14$
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:  $2, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{10} - \frac{1}{6} a^{8} - \frac{1}{12} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} + \frac{1}{6} a^{2} + \frac{1}{3}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{11} - \frac{1}{6} a^{9} - \frac{1}{12} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{3} + \frac{1}{3} a$, $\frac{1}{12} a^{14} - \frac{1}{2} a^{7} + \frac{1}{12} a^{6} - \frac{1}{2} a^{5} + \frac{1}{12} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3}$, $\frac{1}{12} a^{15} + \frac{1}{12} a^{7} + \frac{1}{12} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{78905652} a^{16} - \frac{963519}{26301884} a^{14} + \frac{467182}{19726413} a^{12} - \frac{933139}{11272236} a^{10} - \frac{2567128}{19726413} a^{8} - \frac{1}{2} a^{7} - \frac{6229865}{13150942} a^{6} + \frac{93631}{5636118} a^{4} - \frac{1}{2} a^{3} + \frac{184519}{13150942} a^{2} + \frac{4496284}{19726413}$, $\frac{1}{78905652} a^{17} - \frac{963519}{26301884} a^{15} + \frac{467182}{19726413} a^{13} - \frac{933139}{11272236} a^{11} - \frac{2567128}{19726413} a^{9} - \frac{6229865}{13150942} a^{7} - \frac{1}{2} a^{6} + \frac{93631}{5636118} a^{5} - \frac{1}{2} a^{4} + \frac{184519}{13150942} a^{3} - \frac{1}{2} a^{2} + \frac{4496284}{19726413} a$

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:  \( 19772.5679745 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-2}) \), 3.1.2888.1 x3, 3.3.361.1, 6.0.66724352.2, 6.0.184832.1 x2, 6.0.66724352.1, 9.3.24087491072.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.184832.1
Degree 9 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
$19$19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$