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Properties

Label 18.0.66946260499...0000.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{21}\cdot 5^{12}$
Root discriminant $21.07$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![400, -1800, 4500, -6720, 6450, -4440, 3109, -3177, 3465, -2924, 1821, -969, 628, -483, 309, -142, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 142*x^15 + 309*x^14 - 483*x^13 + 628*x^12 - 969*x^11 + 1821*x^10 - 2924*x^9 + 3465*x^8 - 3177*x^7 + 3109*x^6 - 4440*x^5 + 6450*x^4 - 6720*x^3 + 4500*x^2 - 1800*x + 400)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 142*x^15 + 309*x^14 - 483*x^13 + 628*x^12 - 969*x^11 + 1821*x^10 - 2924*x^9 + 3465*x^8 - 3177*x^7 + 3109*x^6 - 4440*x^5 + 6450*x^4 - 6720*x^3 + 4500*x^2 - 1800*x + 400, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 142 x^{15} + 309 x^{14} - 483 x^{13} + 628 x^{12} - 969 x^{11} + 1821 x^{10} - 2924 x^{9} + 3465 x^{8} - 3177 x^{7} + 3109 x^{6} - 4440 x^{5} + 6450 x^{4} - 6720 x^{3} + 4500 x^{2} - 1800 x + 400 \)

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:  \(-669462604992000000000000=-\,2^{18}\cdot 3^{21}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.07$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{9} - \frac{1}{3} a^{6} - \frac{1}{2} a^{3} + \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{3} a^{7} - \frac{1}{2} a^{4} + \frac{1}{3} a$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{8} - \frac{1}{2} a^{5} - \frac{1}{6} a^{2}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{13} - \frac{1}{12} a^{11} - \frac{1}{12} a^{10} + \frac{1}{12} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{12} a^{5} + \frac{1}{12} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{180} a^{15} - \frac{1}{45} a^{14} - \frac{1}{36} a^{13} - \frac{7}{180} a^{12} - \frac{1}{30} a^{11} + \frac{7}{180} a^{10} - \frac{7}{180} a^{9} - \frac{11}{45} a^{8} - \frac{1}{20} a^{7} - \frac{1}{20} a^{6} - \frac{4}{9} a^{5} + \frac{23}{180} a^{4} + \frac{7}{90} a^{3} - \frac{7}{18} a^{2} + \frac{1}{3} a - \frac{2}{9}$, $\frac{1}{360} a^{16} + \frac{1}{40} a^{14} + \frac{1}{120} a^{13} - \frac{1}{90} a^{12} + \frac{13}{360} a^{11} + \frac{7}{120} a^{10} - \frac{1}{30} a^{9} + \frac{17}{72} a^{8} + \frac{7}{24} a^{7} + \frac{47}{180} a^{6} + \frac{11}{120} a^{5} - \frac{7}{180} a^{4} - \frac{67}{180} a^{3} + \frac{7}{18} a^{2} + \frac{7}{18} a + \frac{2}{9}$, $\frac{1}{4396938508969200} a^{17} + \frac{1424201874043}{2198469254484600} a^{16} + \frac{1660733055053}{879387701793840} a^{15} + \frac{65548445565173}{4396938508969200} a^{14} - \frac{48167870962361}{732823084828200} a^{13} + \frac{142906499096197}{4396938508969200} a^{12} + \frac{61814277008363}{4396938508969200} a^{11} - \frac{13594995431137}{2198469254484600} a^{10} + \frac{24662256311069}{488548723218800} a^{9} - \frac{329241974352103}{1465646169656400} a^{8} + \frac{32197245950}{477928098801} a^{7} + \frac{1543593933578213}{4396938508969200} a^{6} + \frac{96199197293023}{549617313621150} a^{5} + \frac{198028139696309}{439693850896920} a^{4} - \frac{1078881346023}{12213718080470} a^{3} - \frac{32237528147843}{219846925448460} a^{2} + \frac{380258034385}{3664115424141} a - \frac{882573868486}{3664115424141}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{931982613791}{1465646169656400} a^{17} - \frac{623588760583}{732823084828200} a^{16} + \frac{5739307485929}{293129233931280} a^{15} - \frac{58560546555581}{488548723218800} a^{14} + \frac{256099949575793}{732823084828200} a^{13} - \frac{962604084261287}{1465646169656400} a^{12} + \frac{1142122390823687}{1465646169656400} a^{11} - \frac{655364880039193}{732823084828200} a^{10} + \frac{2857378591675849}{1465646169656400} a^{9} - \frac{5799630527643721}{1465646169656400} a^{8} + \frac{1368288400843}{265515610445} a^{7} - \frac{2222513497230461}{488548723218800} a^{6} + \frac{1215414087157159}{366411542414100} a^{5} - \frac{49336876857883}{9770974464376} a^{4} + \frac{55505273526812}{6106859040235} a^{3} - \frac{271419275116749}{24427436160940} a^{2} + \frac{27288772087961}{3664115424141} a - \frac{7501793091055}{3664115424141} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 396553.69170384057 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.200.1, 3.1.675.1 x3, 6.0.1080000.2, 6.0.1366875.1, 9.1.157464000000.1 x3

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

Sibling fields

Galois closure: data not computed
Degree 6 sibling: data not computed
Degree 9 sibling: data not computed
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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
3Data not computed
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$