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Properties

Label 18.18.2474971118...9792.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{18}\cdot 3^{33}\cdot 19^{8}$
Root discriminant $55.47$
Ramified primes $2, 3, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times He_3$ (as 18T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-27, 0, 3078, 0, -21627, 0, 58644, 0, -74196, 0, 43200, 0, -10116, 0, 1089, 0, -54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 54*x^16 + 1089*x^14 - 10116*x^12 + 43200*x^10 - 74196*x^8 + 58644*x^6 - 21627*x^4 + 3078*x^2 - 27)
 
gp: K = bnfinit(x^18 - 54*x^16 + 1089*x^14 - 10116*x^12 + 43200*x^10 - 74196*x^8 + 58644*x^6 - 21627*x^4 + 3078*x^2 - 27, 1)
 

Normalized defining polynomial

\( x^{18} - 54 x^{16} + 1089 x^{14} - 10116 x^{12} + 43200 x^{10} - 74196 x^{8} + 58644 x^{6} - 21627 x^{4} + 3078 x^{2} - 27 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(24749711184581860025231066529792=2^{18}\cdot 3^{33}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.47$
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:  $2, 3, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{9} a^{10} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{11} - \frac{1}{3} a^{5}$, $\frac{1}{9} a^{12}$, $\frac{1}{27} a^{13} - \frac{1}{9} a^{9} + \frac{1}{9} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{108} a^{14} + \frac{1}{9} a^{8} + \frac{1}{3} a^{2} - \frac{1}{4}$, $\frac{1}{108} a^{15} + \frac{1}{9} a^{9} + \frac{1}{3} a^{3} - \frac{1}{4} a$, $\frac{1}{19844784} a^{16} + \frac{30997}{19844784} a^{14} + \frac{5339}{275622} a^{12} + \frac{58967}{1653732} a^{10} + \frac{122845}{1653732} a^{8} + \frac{17744}{137811} a^{6} + \frac{62547}{183748} a^{4} - \frac{100967}{2204976} a^{2} - \frac{128365}{734992}$, $\frac{1}{19844784} a^{17} + \frac{30997}{19844784} a^{15} - \frac{43823}{2480598} a^{13} + \frac{58967}{1653732} a^{11} - \frac{244651}{1653732} a^{9} + \frac{7295}{413433} a^{7} + \frac{3893}{551244} a^{5} - \frac{278653}{734992} a^{3} + \frac{1084889}{2204976} a$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $17$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 11273350288.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3$ (as 18T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 22 conjugacy class representatives for $C_2\times He_3$
Character table for $C_2\times He_3$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{36})^+\), 9.9.5609891727441.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
$19$19.6.4.2$x^{6} - 19 x^{3} + 722$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.6.4.1$x^{6} + 57 x^{3} + 1444$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$