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Properties

Label 18.18.2899550986...1952.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{18}\cdot 7^{15}\cdot 13^{12}$
Root discriminant $55.96$
Ramified primes $2, 7, 13$
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![-664, 5904, -11472, -18108, 75010, -36518, -89615, 85538, 35668, -56382, -3335, 17302, -1134, -2710, 318, 206, -30, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 30*x^16 + 206*x^15 + 318*x^14 - 2710*x^13 - 1134*x^12 + 17302*x^11 - 3335*x^10 - 56382*x^9 + 35668*x^8 + 85538*x^7 - 89615*x^6 - 36518*x^5 + 75010*x^4 - 18108*x^3 - 11472*x^2 + 5904*x - 664)
 
gp: K = bnfinit(x^18 - 6*x^17 - 30*x^16 + 206*x^15 + 318*x^14 - 2710*x^13 - 1134*x^12 + 17302*x^11 - 3335*x^10 - 56382*x^9 + 35668*x^8 + 85538*x^7 - 89615*x^6 - 36518*x^5 + 75010*x^4 - 18108*x^3 - 11472*x^2 + 5904*x - 664, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 30 x^{16} + 206 x^{15} + 318 x^{14} - 2710 x^{13} - 1134 x^{12} + 17302 x^{11} - 3335 x^{10} - 56382 x^{9} + 35668 x^{8} + 85538 x^{7} - 89615 x^{6} - 36518 x^{5} + 75010 x^{4} - 18108 x^{3} - 11472 x^{2} + 5904 x - 664 \)

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:  \(28995509861185340166459512061952=2^{18}\cdot 7^{15}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.96$
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:  $2, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(364=2^{2}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{364}(1,·)$, $\chi_{364}(131,·)$, $\chi_{364}(261,·)$, $\chi_{364}(9,·)$, $\chi_{364}(139,·)$, $\chi_{364}(81,·)$, $\chi_{364}(339,·)$, $\chi_{364}(87,·)$, $\chi_{364}(27,·)$, $\chi_{364}(29,·)$, $\chi_{364}(159,·)$, $\chi_{364}(3,·)$, $\chi_{364}(289,·)$, $\chi_{364}(165,·)$, $\chi_{364}(113,·)$, $\chi_{364}(243,·)$, $\chi_{364}(53,·)$, $\chi_{364}(55,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{2} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{12} - \frac{1}{6} a^{10} + \frac{1}{6} a^{9} + \frac{1}{3} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{15} + \frac{1}{6} a^{12} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{346649172} a^{16} - \frac{2363795}{28887431} a^{15} - \frac{1993519}{173324586} a^{14} - \frac{9670151}{173324586} a^{13} + \frac{9600349}{173324586} a^{12} - \frac{38643157}{173324586} a^{11} - \frac{14535287}{86662293} a^{10} - \frac{21785807}{173324586} a^{9} - \frac{34307885}{346649172} a^{8} - \frac{11776557}{28887431} a^{7} - \frac{29717441}{173324586} a^{6} + \frac{27172157}{173324586} a^{5} - \frac{112500707}{346649172} a^{4} - \frac{32221526}{86662293} a^{3} - \frac{29275142}{86662293} a^{2} - \frac{25801805}{86662293} a + \frac{6732626}{28887431}$, $\frac{1}{877019978615796} a^{17} - \frac{134837}{877019978615796} a^{16} + \frac{7509556134469}{219254994653949} a^{15} + \frac{2281767164860}{73084998217983} a^{14} - \frac{3780472648323}{48723332145322} a^{13} + \frac{53661117000121}{438509989307898} a^{12} + \frac{45691453406905}{438509989307898} a^{11} + \frac{2761650963294}{24361666072661} a^{10} + \frac{144600299944897}{877019978615796} a^{9} + \frac{151824217766173}{877019978615796} a^{8} + \frac{1380970851589}{438509989307898} a^{7} + \frac{33984927596209}{219254994653949} a^{6} + \frac{108677817821405}{877019978615796} a^{5} - \frac{145473481330691}{292339992871932} a^{4} + \frac{50043615837688}{219254994653949} a^{3} - \frac{216002250046549}{438509989307898} a^{2} - \frac{53510587382135}{219254994653949} a + \frac{79573740400030}{219254994653949}$

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:  \( 10619522376.9 \) (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

\(\Q(\sqrt{7}) \), 3.3.169.1, 3.3.8281.1, 3.3.8281.2, \(\Q(\zeta_{7})^+\), 6.6.626971072.1, 6.6.30721582528.1, 6.6.30721582528.2, \(\Q(\zeta_{28})^+\), 9.9.567869252041.1

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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
7Data not computed
$13$13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$