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Properties

Label 18.0.41451359947...6208.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{26}\cdot 3^{31}$
Root discriminant $18.05$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -36, 132, -216, 84, 252, -390, 150, 426, -464, -162, 354, -3, -129, 18, 27, -6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 6*x^16 + 27*x^15 + 18*x^14 - 129*x^13 - 3*x^12 + 354*x^11 - 162*x^10 - 464*x^9 + 426*x^8 + 150*x^7 - 390*x^6 + 252*x^5 + 84*x^4 - 216*x^3 + 132*x^2 - 36*x + 4)
 
gp: K = bnfinit(x^18 - 3*x^17 - 6*x^16 + 27*x^15 + 18*x^14 - 129*x^13 - 3*x^12 + 354*x^11 - 162*x^10 - 464*x^9 + 426*x^8 + 150*x^7 - 390*x^6 + 252*x^5 + 84*x^4 - 216*x^3 + 132*x^2 - 36*x + 4, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 6 x^{16} + 27 x^{15} + 18 x^{14} - 129 x^{13} - 3 x^{12} + 354 x^{11} - 162 x^{10} - 464 x^{9} + 426 x^{8} + 150 x^{7} - 390 x^{6} + 252 x^{5} + 84 x^{4} - 216 x^{3} + 132 x^{2} - 36 x + 4 \)

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:  \(-41451359947637504606208=-\,2^{26}\cdot 3^{31}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.05$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{14} a^{15} - \frac{1}{14} a^{14} + \frac{1}{7} a^{13} - \frac{1}{7} a^{12} + \frac{1}{14} a^{11} + \frac{3}{14} a^{10} - \frac{2}{7} a^{9} - \frac{2}{7} a^{8} + \frac{1}{14} a^{7} + \frac{3}{14} a^{6} + \frac{1}{7} a^{5} + \frac{1}{7} a^{3} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{238} a^{16} - \frac{3}{119} a^{15} - \frac{3}{17} a^{14} - \frac{33}{238} a^{13} + \frac{39}{238} a^{12} - \frac{79}{238} a^{11} - \frac{19}{238} a^{10} - \frac{117}{238} a^{9} + \frac{3}{17} a^{8} - \frac{43}{119} a^{7} + \frac{99}{238} a^{6} + \frac{30}{119} a^{5} - \frac{2}{7} a^{4} + \frac{2}{119} a^{3} - \frac{32}{119} a^{2} - \frac{5}{17} a - \frac{5}{119}$, $\frac{1}{9685173675758} a^{17} + \frac{2093318618}{4842586837879} a^{16} - \frac{155228015973}{9685173675758} a^{15} - \frac{2383119769407}{9685173675758} a^{14} + \frac{123474857847}{1383596239394} a^{13} + \frac{759973958645}{4842586837879} a^{12} + \frac{4182205203745}{9685173675758} a^{11} - \frac{2489213685323}{9685173675758} a^{10} + \frac{4676277163443}{9685173675758} a^{9} + \frac{1132457339017}{4842586837879} a^{8} - \frac{2842683667697}{9685173675758} a^{7} - \frac{1038359220419}{9685173675758} a^{6} + \frac{1028802196747}{4842586837879} a^{5} - \frac{1914193135891}{4842586837879} a^{4} + \frac{37584832387}{284858049287} a^{3} + \frac{84528298685}{4842586837879} a^{2} - \frac{1525991999611}{4842586837879} a - \frac{667678476036}{4842586837879}$

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:  \( -\frac{112159473}{390169346} a^{17} + \frac{298357011}{390169346} a^{16} + \frac{773170831}{390169346} a^{15} - \frac{1383174396}{195084673} a^{14} - \frac{2935100103}{390169346} a^{13} + \frac{960745367}{27869239} a^{12} + \frac{2383850487}{195084673} a^{11} - \frac{18962744301}{195084673} a^{10} + \frac{2896017147}{195084673} a^{9} + \frac{53298799815}{390169346} a^{8} - \frac{2183409615}{27869239} a^{7} - \frac{25591216657}{390169346} a^{6} + \frac{2522530224}{27869239} a^{5} - \frac{8851488483}{195084673} a^{4} - \frac{426128644}{11475569} a^{3} + \frac{1379281755}{27869239} a^{2} - \frac{4665195144}{195084673} a + \frac{881662172}{195084673} \) (order $6$)
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:  \( 42016.82501482997 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.216.1, 6.0.139968.1, 6.0.314928.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.12.22.27$x^{12} + 2 x^{6} + 4$$6$$2$$22$$C_6\times S_3$$[3]_{3}^{6}$
3Data not computed