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Properties

Label 18.0.16643479164...9375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{45}\cdot 5^{9}\cdot 19^{16}$
Root discriminant $477.48$
Ramified primes $3, 5, 19$
Class number $486$ (GRH)
Class group $[3, 9, 18]$ (GRH)
Galois group $S_3\times D_9$ (as 18T50)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6068404224, 0, 0, -2173425048, 0, 0, 403028703, 0, 0, 3789702, 0, 0, 430920, 0, 0, -342, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 342*x^15 + 430920*x^12 + 3789702*x^9 + 403028703*x^6 - 2173425048*x^3 + 6068404224)
 
gp: K = bnfinit(x^18 - 342*x^15 + 430920*x^12 + 3789702*x^9 + 403028703*x^6 - 2173425048*x^3 + 6068404224, 1)
 

Normalized defining polynomial

\( x^{18} - 342 x^{15} + 430920 x^{12} + 3789702 x^{9} + 403028703 x^{6} - 2173425048 x^{3} + 6068404224 \)

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:  \(-1664347916402943860452739891731999463197037109375=-\,3^{45}\cdot 5^{9}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $477.48$
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:  $3, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{7} - \frac{1}{2} a$, $\frac{1}{12} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{228} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{228} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{684} a^{11} - \frac{1}{12} a^{5}$, $\frac{1}{9576} a^{12} - \frac{1}{3192} a^{9} + \frac{1}{24} a^{6} - \frac{23}{56} a^{3} + \frac{1}{7}$, $\frac{1}{19152} a^{13} + \frac{13}{6384} a^{10} - \frac{1}{16} a^{7} - \frac{9}{112} a^{4} + \frac{1}{14} a$, $\frac{1}{363888} a^{14} + \frac{5}{19152} a^{11} - \frac{1}{304} a^{8} - \frac{755}{6384} a^{5} + \frac{3}{14} a^{2}$, $\frac{1}{2109768758972983671696} a^{15} - \frac{524094785908849}{37013486999526029328} a^{12} - \frac{15574337950326339}{12337828999842009776} a^{9} - \frac{530218025617426747}{37013486999526029328} a^{6} + \frac{28482758861086099}{162339855261079076} a^{3} - \frac{5532977549067763}{40584963815269769}$, $\frac{1}{50634450215351608120704} a^{16} - \frac{8958825353369375}{444161843994312351936} a^{13} - \frac{14542583449928896}{6940028812411130499} a^{10} - \frac{1614524865986226353}{49351315999368039104} a^{7} - \frac{2825579915175908873}{15584626105063591296} a^{4} - \frac{210478705424800665}{649359421044316304} a$, $\frac{1}{810151203445625729931264} a^{17} + \frac{84887805124820423}{135025200574270954988544} a^{14} + \frac{1060239654916160305}{1776647375977249407744} a^{11} - \frac{3400263273858096189}{789621055989888625664} a^{8} + \frac{100458145491844036831}{1579242111979777251328} a^{5} - \frac{1277283468569034593}{10389750736709060864} a^{2}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}\times C_{9}\times C_{18}$, which has order $486$ (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:  \( 5491590146240000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times D_9$ (as 18T50):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 18 conjugacy class representatives for $S_3\times D_9$
Character table for $S_3\times D_9$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.3.146205.1, 6.0.320638530375.2

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
3Data not computed
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$19.9.8.1$x^{9} + 76$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.9$x^{9} - 4864$$9$$1$$8$$C_9$$[\ ]_{9}$