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Properties

Label 18.0.12077987568...5647.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,367^{9}$
Root discriminant $19.16$
Ramified prime $367$
Class number $1$
Class group Trivial
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![367, 0, -1404, 0, 1882, 0, -1036, 0, 287, 0, 20, 0, -40, 0, 4, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 4*x^14 - 40*x^12 + 20*x^10 + 287*x^8 - 1036*x^6 + 1882*x^4 - 1404*x^2 + 367)
 
gp: K = bnfinit(x^18 + 4*x^14 - 40*x^12 + 20*x^10 + 287*x^8 - 1036*x^6 + 1882*x^4 - 1404*x^2 + 367, 1)
 

Normalized defining polynomial

\( x^{18} + 4 x^{14} - 40 x^{12} + 20 x^{10} + 287 x^{8} - 1036 x^{6} + 1882 x^{4} - 1404 x^{2} + 367 \)

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:  \(-120779875685608537745647=-\,367^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.16$
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:  $367$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{4} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{5} + \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{6} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{7} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{8} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3}$, $\frac{1}{90} a^{14} - \frac{2}{45} a^{12} + \frac{2}{45} a^{10} + \frac{4}{45} a^{8} - \frac{8}{45} a^{6} - \frac{1}{2} a^{5} - \frac{17}{90} a^{4} + \frac{4}{45} a^{2} + \frac{7}{90}$, $\frac{1}{90} a^{15} - \frac{2}{45} a^{13} + \frac{2}{45} a^{11} - \frac{7}{90} a^{9} - \frac{8}{45} a^{7} - \frac{1}{2} a^{6} - \frac{17}{90} a^{5} - \frac{1}{2} a^{4} + \frac{4}{45} a^{3} - \frac{23}{90} a - \frac{1}{2}$, $\frac{1}{2895539130} a^{16} + \frac{60415}{289553913} a^{14} + \frac{62774153}{2895539130} a^{12} + \frac{101671909}{2895539130} a^{10} - \frac{50695604}{482589855} a^{8} + \frac{92667409}{263230830} a^{6} - \frac{1}{2} a^{5} - \frac{102895670}{289553913} a^{4} - \frac{1}{2} a^{3} - \frac{1184037331}{2895539130} a^{2} - \frac{1}{2} a - \frac{369930146}{1447769565}$, $\frac{1}{2895539130} a^{17} + \frac{60415}{289553913} a^{15} + \frac{62774153}{2895539130} a^{13} + \frac{101671909}{2895539130} a^{11} + \frac{59472077}{965179710} a^{9} + \frac{92667409}{263230830} a^{7} - \frac{1}{2} a^{6} - \frac{102895670}{289553913} a^{5} - \frac{1184037331}{2895539130} a^{3} - \frac{1}{2} a^{2} + \frac{112659709}{1447769565} a - \frac{1}{2}$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 21882.4539804 \)
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

\(\Q(\sqrt{-367}) \), 3.1.367.1 x3, 6.0.49430863.1, 9.1.18141126721.1 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
367Data not computed