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Properties

Label 18.18.4657297996...9568.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{18}\cdot 3^{27}\cdot 13^{12}$
Root discriminant $57.46$
Ramified primes $2, 3, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![181, 1170, -8583, 5474, 32055, -37440, -39087, 60252, 17475, -40488, -1767, 13428, -796, -2292, 252, 190, -27, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 27*x^16 + 190*x^15 + 252*x^14 - 2292*x^13 - 796*x^12 + 13428*x^11 - 1767*x^10 - 40488*x^9 + 17475*x^8 + 60252*x^7 - 39087*x^6 - 37440*x^5 + 32055*x^4 + 5474*x^3 - 8583*x^2 + 1170*x + 181)
 
gp: K = bnfinit(x^18 - 6*x^17 - 27*x^16 + 190*x^15 + 252*x^14 - 2292*x^13 - 796*x^12 + 13428*x^11 - 1767*x^10 - 40488*x^9 + 17475*x^8 + 60252*x^7 - 39087*x^6 - 37440*x^5 + 32055*x^4 + 5474*x^3 - 8583*x^2 + 1170*x + 181, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 27 x^{16} + 190 x^{15} + 252 x^{14} - 2292 x^{13} - 796 x^{12} + 13428 x^{11} - 1767 x^{10} - 40488 x^{9} + 17475 x^{8} + 60252 x^{7} - 39087 x^{6} - 37440 x^{5} + 32055 x^{4} + 5474 x^{3} - 8583 x^{2} + 1170 x + 181 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(46572979962512449327252831469568=2^{18}\cdot 3^{27}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.46$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(468=2^{2}\cdot 3^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{468}(1,·)$, $\chi_{468}(131,·)$, $\chi_{468}(133,·)$, $\chi_{468}(263,·)$, $\chi_{468}(419,·)$, $\chi_{468}(217,·)$, $\chi_{468}(347,·)$, $\chi_{468}(157,·)$, $\chi_{468}(287,·)$, $\chi_{468}(289,·)$, $\chi_{468}(35,·)$, $\chi_{468}(107,·)$, $\chi_{468}(61,·)$, $\chi_{468}(373,·)$, $\chi_{468}(313,·)$, $\chi_{468}(443,·)$, $\chi_{468}(445,·)$, $\chi_{468}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a$, $\frac{1}{39010524908} a^{16} + \frac{834329549}{9752631227} a^{15} - \frac{289138815}{9752631227} a^{14} + \frac{1697494821}{19505262454} a^{13} + \frac{5637132}{9752631227} a^{12} + \frac{2416155741}{19505262454} a^{11} + \frac{1383688878}{9752631227} a^{10} + \frac{791155981}{9752631227} a^{9} + \frac{6478388935}{39010524908} a^{8} + \frac{2128506880}{9752631227} a^{7} - \frac{5312861147}{19505262454} a^{6} + \frac{900486255}{19505262454} a^{5} - \frac{18541662315}{39010524908} a^{4} - \frac{1590450069}{9752631227} a^{3} + \frac{1849456799}{19505262454} a^{2} - \frac{6929035897}{19505262454} a + \frac{3494656147}{39010524908}$, $\frac{1}{17453659938563372} a^{17} + \frac{92409}{17453659938563372} a^{16} - \frac{822533555064261}{4363414984640843} a^{15} + \frac{909925091225513}{4363414984640843} a^{14} - \frac{182267892537635}{4363414984640843} a^{13} + \frac{711308983658403}{4363414984640843} a^{12} - \frac{319308055547651}{8726829969281686} a^{11} - \frac{2109274278661427}{8726829969281686} a^{10} - \frac{2764877220777717}{17453659938563372} a^{9} - \frac{3945914508837501}{17453659938563372} a^{8} + \frac{1359389029907641}{4363414984640843} a^{7} - \frac{1247278694807258}{4363414984640843} a^{6} - \frac{774300805731609}{17453659938563372} a^{5} - \frac{111953959848717}{17453659938563372} a^{4} - \frac{499876232012653}{4363414984640843} a^{3} - \frac{1777782839265070}{4363414984640843} a^{2} + \frac{4303096601014443}{17453659938563372} a - \frac{2810580306773005}{17453659938563372}$

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:  $17$
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:  \( 13132231958.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{3}) \), 3.3.13689.2, 3.3.13689.1, 3.3.169.1, \(\Q(\zeta_{9})^+\), 6.6.35978634432.1, 6.6.35978634432.2, 6.6.49353408.1, \(\Q(\zeta_{36})^+\), 9.9.2565164201769.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
$13$13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$