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Properties

Label 18.0.13706808488...5936.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 137^{9}$
Root discriminant $32.18$
Ramified primes $2, 3, 137$
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![1296, 6480, -1728, -18792, 35316, -39744, 30849, -21345, 12775, -4236, -35, 541, 94, -371, 301, -140, 43, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 43*x^16 - 140*x^15 + 301*x^14 - 371*x^13 + 94*x^12 + 541*x^11 - 35*x^10 - 4236*x^9 + 12775*x^8 - 21345*x^7 + 30849*x^6 - 39744*x^5 + 35316*x^4 - 18792*x^3 - 1728*x^2 + 6480*x + 1296)
 
gp: K = bnfinit(x^18 - 9*x^17 + 43*x^16 - 140*x^15 + 301*x^14 - 371*x^13 + 94*x^12 + 541*x^11 - 35*x^10 - 4236*x^9 + 12775*x^8 - 21345*x^7 + 30849*x^6 - 39744*x^5 + 35316*x^4 - 18792*x^3 - 1728*x^2 + 6480*x + 1296, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 43 x^{16} - 140 x^{15} + 301 x^{14} - 371 x^{13} + 94 x^{12} + 541 x^{11} - 35 x^{10} - 4236 x^{9} + 12775 x^{8} - 21345 x^{7} + 30849 x^{6} - 39744 x^{5} + 35316 x^{4} - 18792 x^{3} - 1728 x^{2} + 6480 x + 1296 \)

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:  \(-1370680848838155021841575936=-\,2^{12}\cdot 3^{9}\cdot 137^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.18$
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:  $2, 3, 137$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{2}$, $\frac{1}{18} a^{8} - \frac{1}{18} a^{7} + \frac{1}{18} a^{6} + \frac{1}{9} a^{5} + \frac{1}{18} a^{4} + \frac{5}{18} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a$, $\frac{1}{18} a^{9} + \frac{1}{3} a^{4} + \frac{5}{18} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{108} a^{10} + \frac{1}{108} a^{9} + \frac{1}{54} a^{8} + \frac{1}{27} a^{7} + \frac{2}{27} a^{6} + \frac{1}{27} a^{5} - \frac{41}{108} a^{4} + \frac{1}{4} a^{3} + \frac{5}{18} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{108} a^{11} + \frac{1}{108} a^{9} + \frac{1}{54} a^{8} + \frac{1}{27} a^{7} - \frac{1}{27} a^{6} - \frac{1}{12} a^{5} + \frac{8}{27} a^{4} - \frac{11}{36} a^{3} - \frac{5}{18} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{216} a^{12} + \frac{1}{216} a^{9} + \frac{1}{108} a^{8} - \frac{1}{27} a^{7} - \frac{17}{216} a^{6} - \frac{1}{27} a^{5} + \frac{11}{54} a^{4} + \frac{29}{72} a^{3} - \frac{17}{36} a^{2} - \frac{1}{6}$, $\frac{1}{216} a^{13} - \frac{1}{216} a^{10} - \frac{1}{216} a^{7} - \frac{1}{18} a^{6} + \frac{1}{9} a^{5} - \frac{71}{216} a^{4} - \frac{4}{9} a^{3} - \frac{5}{18} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{3888} a^{14} - \frac{7}{3888} a^{13} - \frac{1}{486} a^{12} - \frac{5}{3888} a^{11} - \frac{13}{3888} a^{10} - \frac{1}{486} a^{9} - \frac{41}{3888} a^{8} + \frac{83}{3888} a^{7} - \frac{41}{972} a^{6} - \frac{1}{16} a^{5} + \frac{23}{144} a^{4} + \frac{1}{12} a^{3} - \frac{17}{36} a + \frac{1}{18}$, $\frac{1}{3888} a^{15} - \frac{1}{1296} a^{13} - \frac{7}{3888} a^{12} - \frac{1}{324} a^{11} - \frac{1}{432} a^{10} - \frac{79}{3888} a^{9} + \frac{1}{81} a^{8} + \frac{73}{1296} a^{7} - \frac{221}{3888} a^{6} - \frac{11}{108} a^{5} + \frac{7}{144} a^{4} - \frac{17}{72} a^{3} + \frac{5}{18} a^{2} + \frac{1}{4} a - \frac{1}{9}$, $\frac{1}{45077472} a^{16} - \frac{1}{5634684} a^{15} - \frac{767}{45077472} a^{14} + \frac{5509}{45077472} a^{13} - \frac{2719}{1252152} a^{12} + \frac{97939}{45077472} a^{11} - \frac{168635}{45077472} a^{10} + \frac{35119}{1878228} a^{9} - \frac{651545}{45077472} a^{8} + \frac{332311}{5008608} a^{7} - \frac{130873}{11269368} a^{6} + \frac{27403}{185504} a^{5} + \frac{176677}{834768} a^{4} + \frac{17569}{104346} a^{3} + \frac{103045}{417384} a^{2} - \frac{2840}{17391} a - \frac{12877}{52173}$, $\frac{1}{21051179424} a^{17} + \frac{25}{2339019936} a^{16} - \frac{1370723}{21051179424} a^{15} - \frac{185327}{5262794856} a^{14} + \frac{24918631}{21051179424} a^{13} + \frac{3516595}{21051179424} a^{12} - \frac{37479367}{10525589712} a^{11} - \frac{55271993}{21051179424} a^{10} - \frac{17682661}{1238304672} a^{9} - \frac{11027285}{877132476} a^{8} + \frac{1206248389}{21051179424} a^{7} - \frac{211428445}{7017059808} a^{6} - \frac{213195971}{2339019936} a^{5} - \frac{96396979}{194918328} a^{4} - \frac{26037845}{194918328} a^{3} - \frac{9016299}{21657592} a^{2} - \frac{24310703}{97459164} a - \frac{7412693}{16243194}$

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:  \( 28638652.8041 \) (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

\(\Q(\sqrt{-411}) \), 3.1.411.1 x3, 6.0.69426531.1, 9.1.1826195471424.1 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 R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$$[\ ]_{3}^{2}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$137$137.2.1.2$x^{2} + 411$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.1.2$x^{2} + 411$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.1.2$x^{2} + 411$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.1.2$x^{2} + 411$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.1.2$x^{2} + 411$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.1.2$x^{2} + 411$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.1.2$x^{2} + 411$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.1.2$x^{2} + 411$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.1.2$x^{2} + 411$$2$$1$$1$$C_2$$[\ ]_{2}$