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Properties

Label 18.0.19534931466...5616.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 331^{9}$
Root discriminant $28.88$
Ramified primes $2, 331$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $D_9$ (as 18T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![441, -1113, 5710, -27701, 67479, -92740, 79963, -47101, 22487, -11370, 6609, -4023, 2177, -916, 325, -103, 24, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 24*x^16 - 103*x^15 + 325*x^14 - 916*x^13 + 2177*x^12 - 4023*x^11 + 6609*x^10 - 11370*x^9 + 22487*x^8 - 47101*x^7 + 79963*x^6 - 92740*x^5 + 67479*x^4 - 27701*x^3 + 5710*x^2 - 1113*x + 441)
 
gp: K = bnfinit(x^18 - 5*x^17 + 24*x^16 - 103*x^15 + 325*x^14 - 916*x^13 + 2177*x^12 - 4023*x^11 + 6609*x^10 - 11370*x^9 + 22487*x^8 - 47101*x^7 + 79963*x^6 - 92740*x^5 + 67479*x^4 - 27701*x^3 + 5710*x^2 - 1113*x + 441, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} + 24 x^{16} - 103 x^{15} + 325 x^{14} - 916 x^{13} + 2177 x^{12} - 4023 x^{11} + 6609 x^{10} - 11370 x^{9} + 22487 x^{8} - 47101 x^{7} + 79963 x^{6} - 92740 x^{5} + 67479 x^{4} - 27701 x^{3} + 5710 x^{2} - 1113 x + 441 \)

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:  \(-195349314665489411651055616=-\,2^{12}\cdot 331^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.88$
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:  $2, 331$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{12} a^{10} - \frac{1}{12} a^{9} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{3} a^{2} - \frac{5}{12} a + \frac{1}{4}$, $\frac{1}{36} a^{11} + \frac{1}{36} a^{10} - \frac{1}{36} a^{8} - \frac{11}{36} a^{7} - \frac{4}{9} a^{6} - \frac{5}{36} a^{5} + \frac{11}{36} a^{4} + \frac{1}{3} a^{3} - \frac{5}{36} a^{2} - \frac{11}{36} a - \frac{1}{3}$, $\frac{1}{72} a^{12} - \frac{1}{72} a^{11} + \frac{1}{72} a^{10} - \frac{1}{18} a^{9} + \frac{1}{24} a^{8} - \frac{11}{24} a^{7} + \frac{1}{6} a^{6} + \frac{1}{8} a^{5} + \frac{11}{72} a^{4} - \frac{7}{36} a^{3} - \frac{1}{72} a^{2} + \frac{19}{72} a + \frac{11}{24}$, $\frac{1}{72} a^{13} - \frac{1}{24} a^{10} - \frac{1}{72} a^{9} - \frac{1}{12} a^{8} + \frac{3}{8} a^{7} - \frac{3}{8} a^{6} - \frac{1}{18} a^{5} + \frac{7}{24} a^{4} + \frac{11}{24} a^{3} - \frac{5}{12} a^{2} + \frac{7}{18} a + \frac{11}{24}$, $\frac{1}{432} a^{14} + \frac{1}{432} a^{13} - \frac{1}{216} a^{12} + \frac{1}{144} a^{11} - \frac{7}{216} a^{10} - \frac{23}{432} a^{9} + \frac{11}{432} a^{8} - \frac{79}{216} a^{7} + \frac{25}{432} a^{6} + \frac{65}{144} a^{5} + \frac{5}{54} a^{4} + \frac{7}{432} a^{3} - \frac{29}{108} a^{2} - \frac{35}{144} a - \frac{17}{48}$, $\frac{1}{864} a^{15} - \frac{1}{288} a^{13} - \frac{1}{864} a^{12} + \frac{1}{864} a^{11} - \frac{1}{288} a^{10} - \frac{7}{432} a^{9} - \frac{55}{864} a^{8} + \frac{35}{288} a^{7} - \frac{83}{432} a^{6} - \frac{413}{864} a^{5} - \frac{85}{288} a^{4} - \frac{85}{288} a^{3} + \frac{101}{864} a^{2} + \frac{47}{144} a - \frac{15}{32}$, $\frac{1}{18144} a^{16} + \frac{1}{2592} a^{15} + \frac{17}{18144} a^{14} + \frac{47}{9072} a^{13} - \frac{47}{9072} a^{12} + \frac{1}{162} a^{11} - \frac{7}{864} a^{10} + \frac{275}{18144} a^{9} - \frac{233}{1512} a^{8} + \frac{3385}{18144} a^{7} + \frac{5189}{18144} a^{6} + \frac{3569}{9072} a^{5} + \frac{208}{567} a^{4} - \frac{277}{567} a^{3} + \frac{2173}{18144} a^{2} + \frac{247}{6048} a + \frac{127}{288}$, $\frac{1}{374958280948724928} a^{17} + \frac{1070167662205}{46869785118590616} a^{16} + \frac{16203048243209}{31246523412393744} a^{15} + \frac{57763497535813}{124986093649574976} a^{14} - \frac{80999297742191}{15623261706196872} a^{13} + \frac{359186389028993}{62493046824787488} a^{12} - \frac{707827622589761}{53565468706960704} a^{11} + \frac{51931093384085}{1802684043022716} a^{10} - \frac{2262116082516163}{53565468706960704} a^{9} + \frac{48002808807796657}{374958280948724928} a^{8} - \frac{28572450326939023}{62493046824787488} a^{7} + \frac{12035898482892715}{41662031216524992} a^{6} - \frac{22192828209559535}{62493046824787488} a^{5} - \frac{33477748078987}{85842097286796} a^{4} + \frac{12181551915328307}{53565468706960704} a^{3} + \frac{523417429258637}{26782734353480352} a^{2} + \frac{20718064685454953}{62493046824787488} a - \frac{2791804594347209}{5951718745217856}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( -1 \) (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:  \( 449668.225814 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9$ (as 18T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $D_9$
Character table for $D_9$

Intermediate fields

\(\Q(\sqrt{-331}) \), 3.1.331.1 x3, 6.0.36264691.1, 9.1.768231214144.1 x9

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

Sibling fields

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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
331Data not computed