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Properties

Label 18.0.31619050548...2896.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 11^{9}\cdot 41^{9}$
Root discriminant $33.71$
Ramified primes $2, 11, 41$
Class number $2$ (GRH)
Class group $[2]$ (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![1804, 9020, 65288, -106788, 165508, -187176, 166209, -123691, 73403, -34858, 14283, -5017, 1616, -515, 147, -36, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 9*x^16 - 36*x^15 + 147*x^14 - 515*x^13 + 1616*x^12 - 5017*x^11 + 14283*x^10 - 34858*x^9 + 73403*x^8 - 123691*x^7 + 166209*x^6 - 187176*x^5 + 165508*x^4 - 106788*x^3 + 65288*x^2 + 9020*x + 1804)
 
gp: K = bnfinit(x^18 - 3*x^17 + 9*x^16 - 36*x^15 + 147*x^14 - 515*x^13 + 1616*x^12 - 5017*x^11 + 14283*x^10 - 34858*x^9 + 73403*x^8 - 123691*x^7 + 166209*x^6 - 187176*x^5 + 165508*x^4 - 106788*x^3 + 65288*x^2 + 9020*x + 1804, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 9 x^{16} - 36 x^{15} + 147 x^{14} - 515 x^{13} + 1616 x^{12} - 5017 x^{11} + 14283 x^{10} - 34858 x^{9} + 73403 x^{8} - 123691 x^{7} + 166209 x^{6} - 187176 x^{5} + 165508 x^{4} - 106788 x^{3} + 65288 x^{2} + 9020 x + 1804 \)

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:  \(-3161905054840233264308432896=-\,2^{12}\cdot 11^{9}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.71$
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:  $2, 11, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{2}{9} a^{5} - \frac{1}{3} a^{4} + \frac{2}{9} a^{3} + \frac{2}{9}$, $\frac{1}{9} a^{10} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{2}{9} a^{4} - \frac{2}{9} a^{3} + \frac{1}{3} a^{2} + \frac{2}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{2}{9} a^{5} + \frac{1}{9} a^{4} - \frac{2}{9} a^{3} + \frac{2}{9} a^{2} + \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{198} a^{12} + \frac{1}{99} a^{11} + \frac{5}{99} a^{10} - \frac{7}{198} a^{9} - \frac{2}{99} a^{8} + \frac{4}{99} a^{7} - \frac{7}{66} a^{6} + \frac{26}{99} a^{5} + \frac{43}{99} a^{4} - \frac{1}{6} a^{3} + \frac{19}{99} a^{2} - \frac{1}{9} a + \frac{4}{9}$, $\frac{1}{594} a^{13} - \frac{8}{297} a^{11} - \frac{1}{22} a^{10} + \frac{16}{297} a^{9} + \frac{19}{297} a^{8} - \frac{37}{594} a^{7} + \frac{25}{297} a^{6} + \frac{10}{33} a^{5} + \frac{169}{594} a^{4} - \frac{1}{99} a^{3} - \frac{104}{297} a^{2} - \frac{7}{27} a - \frac{1}{27}$, $\frac{1}{1188} a^{14} - \frac{1}{1188} a^{12} + \frac{1}{396} a^{11} - \frac{4}{297} a^{10} - \frac{1}{1188} a^{9} - \frac{31}{1188} a^{8} - \frac{47}{594} a^{7} + \frac{7}{132} a^{6} - \frac{173}{1188} a^{5} - \frac{25}{99} a^{4} + \frac{419}{1188} a^{3} + \frac{109}{594} a^{2} - \frac{7}{54} a + \frac{1}{18}$, $\frac{1}{1188} a^{15} - \frac{1}{1188} a^{13} - \frac{1}{396} a^{12} - \frac{7}{297} a^{11} - \frac{61}{1188} a^{10} + \frac{1}{108} a^{9} - \frac{35}{594} a^{8} + \frac{5}{396} a^{7} - \frac{47}{1188} a^{6} + \frac{16}{33} a^{5} - \frac{97}{1188} a^{4} + \frac{104}{297} a^{3} - \frac{191}{594} a^{2} + \frac{1}{6} a - \frac{4}{9}$, $\frac{1}{1293732} a^{16} + \frac{133}{323433} a^{15} - \frac{5}{117612} a^{14} + \frac{3}{5324} a^{13} + \frac{614}{323433} a^{12} + \frac{1865}{39204} a^{11} - \frac{18581}{1293732} a^{10} + \frac{3919}{215622} a^{9} + \frac{5881}{1293732} a^{8} + \frac{4049}{39204} a^{7} + \frac{1516}{35937} a^{6} - \frac{280747}{1293732} a^{5} - \frac{63814}{323433} a^{4} - \frac{209065}{646866} a^{3} - \frac{154507}{646866} a^{2} - \frac{8231}{29403} a + \frac{5986}{29403}$, $\frac{1}{70121961960491008414836} a^{17} - \frac{7969733014294795}{23373987320163669471612} a^{16} - \frac{13306844417723556899}{70121961960491008414836} a^{15} - \frac{10303118609984772503}{70121961960491008414836} a^{14} - \frac{15270090614872103071}{70121961960491008414836} a^{13} + \frac{13584267385915484011}{17530490490122752103709} a^{12} + \frac{995620583472165833653}{70121961960491008414836} a^{11} - \frac{42067287996845962495}{1149540360008049318276} a^{10} + \frac{819200024812067236793}{35060980980245504207418} a^{9} - \frac{9276982766979911965651}{70121961960491008414836} a^{8} - \frac{2003463203460366686227}{23373987320163669471612} a^{7} - \frac{1922934167629039377664}{17530490490122752103709} a^{6} - \frac{353671768551564642965}{11686993660081834735806} a^{5} - \frac{5942833640442873669766}{17530490490122752103709} a^{4} + \frac{11182535442251883888245}{23373987320163669471612} a^{3} + \frac{7432648733778722483839}{17530490490122752103709} a^{2} + \frac{47392476896256457519}{96586724463486237486} a + \frac{474837439213720361305}{3187361907295045837038}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 7560775.8741 \) (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

\(\Q(\sqrt{-451}) \), 3.1.451.1 x3, 6.0.91733851.1, 9.1.2647805875264.1 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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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$$[\ ]_{3}^{2}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$