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Properties

Label 18.0.58433528716...7867.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 19^{6}\cdot 739^{6}$
Root discriminant $125.36$
Ramified primes $3, 19, 739$
Class number $108$ (GRH)
Class group $[3, 6, 6]$ (GRH)
Galois group $C_3^2\times S_3$ (as 18T17)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![975896298559, 0, 0, -50173783396, 0, 0, 1155284731, 0, 0, -15206342, 0, 0, 119023, 0, 0, -519, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 519*x^15 + 119023*x^12 - 15206342*x^9 + 1155284731*x^6 - 50173783396*x^3 + 975896298559)
 
gp: K = bnfinit(x^18 - 519*x^15 + 119023*x^12 - 15206342*x^9 + 1155284731*x^6 - 50173783396*x^3 + 975896298559, 1)
 

Normalized defining polynomial

\( x^{18} - 519 x^{15} + 119023 x^{12} - 15206342 x^{9} + 1155284731 x^{6} - 50173783396 x^{3} + 975896298559 \)

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:  \(-58433528716150185268173747532031037867=-\,3^{27}\cdot 19^{6}\cdot 739^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $125.36$
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:  $3, 19, 739$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{682614728680440329513581} a^{15} + \frac{34476831071612114843531}{682614728680440329513581} a^{12} + \frac{239848892034221967798346}{682614728680440329513581} a^{9} + \frac{175990825581497156342192}{682614728680440329513581} a^{6} - \frac{279647901085697651258894}{682614728680440329513581} a^{3} + \frac{215613482884533664110141}{682614728680440329513581}$, $\frac{1}{6770855493781287628445209939} a^{16} - \frac{138536313091057774776413412}{6770855493781287628445209939} a^{13} + \frac{1851490993073388395608630018}{6770855493781287628445209939} a^{10} + \frac{239091145863735612486095542}{6770855493781287628445209939} a^{7} + \frac{2649630728836383661520462548}{6770855493781287628445209939} a^{4} - \frac{2526824112092105566195166721}{6770855493781287628445209939} a$, $\frac{1}{67160115642816591986548037384941} a^{17} + \frac{25390569565366737548894760857838}{67160115642816591986548037384941} a^{14} + \frac{29746219675174269940155415892045}{67160115642816591986548037384941} a^{11} - \frac{11733653479577107724483062728745}{67160115642816591986548037384941} a^{8} - \frac{20851585290117529511949726149572}{67160115642816591986548037384941} a^{5} + \frac{25821516029169738909323835540625}{67160115642816591986548037384941} a^{2}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{6}$, which has order $108$ (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:  \( -\frac{426221938246262745260}{3534742928569294315081475651839} a^{17} + \frac{180139603504639304663412}{3534742928569294315081475651839} a^{14} - \frac{32860332885956221366084816}{3534742928569294315081475651839} a^{11} + \frac{3149608046671258531011521541}{3534742928569294315081475651839} a^{8} - \frac{167291850270733017453148684374}{3534742928569294315081475651839} a^{5} + \frac{4041177607590361091263661047129}{3534742928569294315081475651839} a^{2} \) (order $18$)
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:  \( 22671106884.393993 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2\times S_3$ (as 18T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 27 conjugacy class representatives for $C_3^2\times S_3$
Character table for $C_3^2\times S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), 6.0.5323041387.4, Deg 6, 6.0.3880497171123.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
739Data not computed