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Properties

Label 18.10.1217816758...9257.2
Degree $18$
Signature $[10, 4]$
Discriminant $3^{24}\cdot 73^{3}\cdot 577^{4}$
Root discriminant $36.33$
Ramified primes $3, 73, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T765

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2809, -14151, -37893, -69436, -83958, -58506, -6894, 31575, 32964, 12397, -3936, -6645, -1824, 870, 453, -35, -36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 36*x^16 - 35*x^15 + 453*x^14 + 870*x^13 - 1824*x^12 - 6645*x^11 - 3936*x^10 + 12397*x^9 + 32964*x^8 + 31575*x^7 - 6894*x^6 - 58506*x^5 - 83958*x^4 - 69436*x^3 - 37893*x^2 - 14151*x - 2809)
 
gp: K = bnfinit(x^18 - 36*x^16 - 35*x^15 + 453*x^14 + 870*x^13 - 1824*x^12 - 6645*x^11 - 3936*x^10 + 12397*x^9 + 32964*x^8 + 31575*x^7 - 6894*x^6 - 58506*x^5 - 83958*x^4 - 69436*x^3 - 37893*x^2 - 14151*x - 2809, 1)
 

Normalized defining polynomial

\( x^{18} - 36 x^{16} - 35 x^{15} + 453 x^{14} + 870 x^{13} - 1824 x^{12} - 6645 x^{11} - 3936 x^{10} + 12397 x^{9} + 32964 x^{8} + 31575 x^{7} - 6894 x^{6} - 58506 x^{5} - 83958 x^{4} - 69436 x^{3} - 37893 x^{2} - 14151 x - 2809 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12178167588536804870357659257=3^{24}\cdot 73^{3}\cdot 577^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.33$
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:  $3, 73, 577$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3034957409858107674043606877573359} a^{17} + \frac{15845382538272755088291294126465}{57263347355813352340445412784403} a^{16} - \frac{70511768729790693607368215795892}{159734600518847772318084572503861} a^{15} + \frac{447196312164962223068794864485404}{3034957409858107674043606877573359} a^{14} - \frac{61031154922732242062116376097070}{3034957409858107674043606877573359} a^{13} + \frac{1063411488186163280130946075725922}{3034957409858107674043606877573359} a^{12} - \frac{538546420785746125257338878184965}{3034957409858107674043606877573359} a^{11} - \frac{898201365198189805259514005219940}{3034957409858107674043606877573359} a^{10} - \frac{779363871671104109388402404780457}{3034957409858107674043606877573359} a^{9} + \frac{1267769445211267252856695748169867}{3034957409858107674043606877573359} a^{8} + \frac{717282928662505289274755707401679}{3034957409858107674043606877573359} a^{7} + \frac{53812512015035729732315485589917}{159734600518847772318084572503861} a^{6} - \frac{1021654140625423495107969250640632}{3034957409858107674043606877573359} a^{5} - \frac{488360427627883123627049977888589}{3034957409858107674043606877573359} a^{4} - \frac{241805294451281955950632845989126}{3034957409858107674043606877573359} a^{3} - \frac{1392354792418210128697414138414798}{3034957409858107674043606877573359} a^{2} - \frac{502586362191254645438937917322351}{3034957409858107674043606877573359} a + \frac{23221805454536724645198865714831}{57263347355813352340445412784403}$

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:  $13$
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:  \( 22875164.9454 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T765:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n765 are not computed
Character table for t18n765 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.22384826361.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 siblings: 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 $18$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$73$$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
73.6.0.1$x^{6} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
577Data not computed