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Properties

Label 18.0.94903974258...8704.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}\cdot 7^{15}$
Root discriminant $52.60$
Ramified primes $2, 3, 7$
Class number $1456$ (GRH)
Class group $[2, 2, 2, 182]$ (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![9261, 0, 55566, 0, 120393, 0, 130536, 0, 79380, 0, 28224, 0, 5880, 0, 693, 0, 42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 42*x^16 + 693*x^14 + 5880*x^12 + 28224*x^10 + 79380*x^8 + 130536*x^6 + 120393*x^4 + 55566*x^2 + 9261)
 
gp: K = bnfinit(x^18 + 42*x^16 + 693*x^14 + 5880*x^12 + 28224*x^10 + 79380*x^8 + 130536*x^6 + 120393*x^4 + 55566*x^2 + 9261, 1)
 

Normalized defining polynomial

\( x^{18} + 42 x^{16} + 693 x^{14} + 5880 x^{12} + 28224 x^{10} + 79380 x^{8} + 130536 x^{6} + 120393 x^{4} + 55566 x^{2} + 9261 \)

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:  \(-9490397425838961457555240648704=-\,2^{18}\cdot 3^{27}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.60$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(252=2^{2}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{252}(1,·)$, $\chi_{252}(131,·)$, $\chi_{252}(193,·)$, $\chi_{252}(205,·)$, $\chi_{252}(143,·)$, $\chi_{252}(83,·)$, $\chi_{252}(85,·)$, $\chi_{252}(215,·)$, $\chi_{252}(25,·)$, $\chi_{252}(227,·)$, $\chi_{252}(37,·)$, $\chi_{252}(167,·)$, $\chi_{252}(169,·)$, $\chi_{252}(109,·)$, $\chi_{252}(47,·)$, $\chi_{252}(251,·)$, $\chi_{252}(121,·)$, $\chi_{252}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{21} a^{6}$, $\frac{1}{21} a^{7}$, $\frac{1}{21} a^{8}$, $\frac{1}{21} a^{9}$, $\frac{1}{21} a^{10}$, $\frac{1}{21} a^{11}$, $\frac{1}{441} a^{12}$, $\frac{1}{441} a^{13}$, $\frac{1}{1764} a^{14} + \frac{1}{4}$, $\frac{1}{1764} a^{15} + \frac{1}{4} a$, $\frac{1}{7056} a^{16} + \frac{1}{7056} a^{14} - \frac{1}{1764} a^{12} - \frac{1}{84} a^{10} + \frac{1}{84} a^{8} - \frac{1}{2} a^{4} - \frac{7}{16} a^{2} - \frac{3}{16}$, $\frac{1}{7056} a^{17} + \frac{1}{7056} a^{15} - \frac{1}{1764} a^{13} - \frac{1}{84} a^{11} + \frac{1}{84} a^{9} - \frac{1}{2} a^{5} - \frac{7}{16} a^{3} - \frac{3}{16} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{182}$, which has order $1456$ (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:  \( 54408.4888887 \) (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

\(\Q(\sqrt{-21}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.1, 3.3.3969.2, 6.0.432081216.1, 6.0.29042496.1, 6.0.21171979584.2, 6.0.21171979584.1, 9.9.62523502209.1

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$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
7Data not computed