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Properties

Label 18.0.19636546131...0967.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{9}\cdot 17^{16}$
Root discriminant $32.83$
Ramified primes $7, 17$
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![567, 0, -1301, 0, 1383, 0, -2286, 0, 3810, 0, 1048, 0, 471, 0, -81, 0, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^16 - 81*x^14 + 471*x^12 + 1048*x^10 + 3810*x^8 - 2286*x^6 + 1383*x^4 - 1301*x^2 + 567)
 
gp: K = bnfinit(x^18 - 3*x^16 - 81*x^14 + 471*x^12 + 1048*x^10 + 3810*x^8 - 2286*x^6 + 1383*x^4 - 1301*x^2 + 567, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{16} - 81 x^{14} + 471 x^{12} + 1048 x^{10} + 3810 x^{8} - 2286 x^{6} + 1383 x^{4} - 1301 x^{2} + 567 \)

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:  \(-1963654613102253673602960967=-\,7^{9}\cdot 17^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.83$
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:  $7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{42} a^{10} - \frac{2}{7} a^{8} + \frac{1}{14} a^{6} - \frac{2}{7} a^{4} - \frac{1}{2} a^{3} + \frac{4}{21} a^{2} - \frac{1}{2}$, $\frac{1}{42} a^{11} + \frac{1}{21} a^{9} + \frac{1}{14} a^{7} - \frac{2}{7} a^{5} - \frac{1}{2} a^{4} + \frac{4}{21} a^{3} + \frac{1}{6} a$, $\frac{1}{210} a^{12} - \frac{1}{105} a^{10} + \frac{31}{70} a^{8} - \frac{11}{35} a^{6} - \frac{1}{2} a^{5} + \frac{4}{15} a^{4} + \frac{101}{210} a^{2} + \frac{2}{5}$, $\frac{1}{210} a^{13} - \frac{1}{105} a^{11} - \frac{2}{35} a^{9} - \frac{1}{2} a^{8} + \frac{13}{70} a^{7} + \frac{4}{15} a^{5} + \frac{101}{210} a^{3} - \frac{1}{2} a^{2} - \frac{1}{10} a - \frac{1}{2}$, $\frac{1}{51870} a^{14} - \frac{1}{5187} a^{12} + \frac{11}{2730} a^{10} + \frac{23}{91} a^{8} - \frac{1}{2} a^{7} - \frac{10268}{25935} a^{6} + \frac{259}{7410} a^{4} - \frac{8992}{25935} a^{2} + \frac{124}{1235}$, $\frac{1}{155610} a^{15} - \frac{1}{15561} a^{13} + \frac{38}{4095} a^{11} + \frac{73}{1638} a^{9} - \frac{21383}{77805} a^{7} - \frac{1}{2} a^{6} - \frac{13007}{155610} a^{5} - \frac{1}{2} a^{4} + \frac{21883}{77805} a^{3} - \frac{1}{2} a^{2} + \frac{5312}{11115} a - \frac{1}{2}$, $\frac{1}{10704256290} a^{16} + \frac{89321}{10704256290} a^{14} - \frac{873406}{764589735} a^{12} - \frac{15473}{112676382} a^{10} - \frac{37180075}{82340433} a^{8} + \frac{386337025}{1070425629} a^{6} - \frac{1}{2} a^{5} + \frac{977480261}{10704256290} a^{4} - \frac{1730035882}{5352128145} a^{2} - \frac{1}{2} a + \frac{46270647}{169908830}$, $\frac{1}{10704256290} a^{17} + \frac{3422}{1784042715} a^{15} - \frac{39251}{36409035} a^{13} - \frac{589481}{62597990} a^{11} + \frac{4166668}{1070425629} a^{9} - \frac{1}{2} a^{8} + \frac{484357493}{3568085430} a^{7} - \frac{1}{2} a^{6} + \frac{312036464}{1784042715} a^{5} - \frac{1}{2} a^{4} + \frac{235198064}{594680905} a^{3} - \frac{1}{2} a^{2} + \frac{225105611}{764589735} a$

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:  \( 10137100.5942 \) (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

\(\Q(\sqrt{-7}) \), 3.1.2023.1 x3, 6.0.28647703.1, 9.1.16748793615841.1 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
17Data not computed