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Properties

Label 18.0.74673302315...8751.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 31^{9}$
Root discriminant $24.09$
Ramified primes $3, 31$
Class number $1$ (GRH)
Class group Trivial (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![5239, 0, 5373, 0, 6012, 0, 4566, 0, 3078, 0, 1575, 0, 570, 0, 135, 0, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 18*x^16 + 135*x^14 + 570*x^12 + 1575*x^10 + 3078*x^8 + 4566*x^6 + 6012*x^4 + 5373*x^2 + 5239)
 
gp: K = bnfinit(x^18 + 18*x^16 + 135*x^14 + 570*x^12 + 1575*x^10 + 3078*x^8 + 4566*x^6 + 6012*x^4 + 5373*x^2 + 5239, 1)
 

Normalized defining polynomial

\( x^{18} + 18 x^{16} + 135 x^{14} + 570 x^{12} + 1575 x^{10} + 3078 x^{8} + 4566 x^{6} + 6012 x^{4} + 5373 x^{2} + 5239 \)

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:  \(-7467330231571086237938751=-\,3^{24}\cdot 31^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.09$
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:  $3, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{9} a^{6} - \frac{1}{3} a^{4} - \frac{2}{9}$, $\frac{1}{9} a^{7} - \frac{1}{3} a^{5} - \frac{2}{9} a$, $\frac{1}{9} a^{8} - \frac{2}{9} a^{2} + \frac{1}{3}$, $\frac{1}{18} a^{9} - \frac{1}{18} a^{7} + \frac{1}{6} a^{5} - \frac{1}{9} a^{3} + \frac{5}{18} a - \frac{1}{2}$, $\frac{1}{18} a^{10} - \frac{1}{18} a^{8} - \frac{1}{18} a^{6} - \frac{4}{9} a^{4} + \frac{5}{18} a^{2} - \frac{1}{2} a + \frac{4}{9}$, $\frac{1}{18} a^{11} + \frac{7}{18} a^{5} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{486} a^{12} + \frac{2}{81} a^{10} + \frac{5}{486} a^{6} - \frac{17}{162} a^{4} - \frac{1}{2} a^{3} + \frac{17}{54} a^{2} - \frac{1}{2} a + \frac{20}{243}$, $\frac{1}{486} a^{13} + \frac{2}{81} a^{11} + \frac{5}{486} a^{7} - \frac{17}{162} a^{5} - \frac{1}{2} a^{4} + \frac{17}{54} a^{3} - \frac{1}{2} a^{2} + \frac{20}{243} a$, $\frac{1}{486} a^{14} - \frac{1}{54} a^{10} - \frac{11}{243} a^{8} + \frac{4}{81} a^{6} - \frac{1}{2} a^{5} - \frac{17}{54} a^{4} - \frac{1}{2} a^{3} + \frac{121}{486} a^{2} - \frac{1}{2} a - \frac{17}{81}$, $\frac{1}{6318} a^{15} + \frac{1}{3159} a^{13} - \frac{11}{1053} a^{11} - \frac{103}{6318} a^{9} - \frac{209}{6318} a^{7} - \frac{1}{18} a^{6} + \frac{35}{162} a^{5} + \frac{1}{6} a^{4} - \frac{70}{3159} a^{3} - \frac{983}{3159} a + \frac{1}{9}$, $\frac{1}{69498} a^{16} + \frac{14}{34749} a^{14} - \frac{7}{34749} a^{12} + \frac{287}{69498} a^{10} + \frac{2729}{69498} a^{8} - \frac{1}{18} a^{7} - \frac{205}{5346} a^{6} + \frac{1}{6} a^{5} + \frac{827}{34749} a^{4} + \frac{16853}{34749} a^{2} + \frac{1}{9} a - \frac{211}{2673}$, $\frac{1}{69498} a^{17} - \frac{5}{69498} a^{15} + \frac{7}{7722} a^{13} + \frac{160}{34749} a^{11} - \frac{797}{34749} a^{9} - \frac{1}{18} a^{8} - \frac{925}{23166} a^{7} + \frac{14953}{69498} a^{5} - \frac{2716}{34749} a^{3} + \frac{1}{9} a^{2} + \frac{7547}{23166} a - \frac{1}{6}$

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:  \( -1 \) (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:  \( 94195.6338059 \) (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

\(\Q(\sqrt{-31}) \), 3.1.31.1 x3, 6.0.29791.1, 9.1.490796923761.1 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$3$3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$