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Properties

Label 18.0.48590834820...6448.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{27}\cdot 7^{15}$
Root discriminant $74.38$
Ramified primes $2, 3, 7$
Class number $40768$ (GRH)
Class group $[2, 2, 28, 364]$ (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![4741632, 0, 14224896, 0, 15410304, 0, 8354304, 0, 2540160, 0, 451584, 0, 47040, 0, 2772, 0, 84, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 84*x^16 + 2772*x^14 + 47040*x^12 + 451584*x^10 + 2540160*x^8 + 8354304*x^6 + 15410304*x^4 + 14224896*x^2 + 4741632)
 
gp: K = bnfinit(x^18 + 84*x^16 + 2772*x^14 + 47040*x^12 + 451584*x^10 + 2540160*x^8 + 8354304*x^6 + 15410304*x^4 + 14224896*x^2 + 4741632, 1)
 

Normalized defining polynomial

\( x^{18} + 84 x^{16} + 2772 x^{14} + 47040 x^{12} + 451584 x^{10} + 2540160 x^{8} + 8354304 x^{6} + 15410304 x^{4} + 14224896 x^{2} + 4741632 \)

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:  \(-4859083482029548266268283212136448=-\,2^{27}\cdot 3^{27}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.38$
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:  \(504=2^{3}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{504}(1,·)$, $\chi_{504}(131,·)$, $\chi_{504}(193,·)$, $\chi_{504}(457,·)$, $\chi_{504}(395,·)$, $\chi_{504}(337,·)$, $\chi_{504}(83,·)$, $\chi_{504}(25,·)$, $\chi_{504}(289,·)$, $\chi_{504}(419,·)$, $\chi_{504}(227,·)$, $\chi_{504}(169,·)$, $\chi_{504}(299,·)$, $\chi_{504}(59,·)$, $\chi_{504}(467,·)$, $\chi_{504}(361,·)$, $\chi_{504}(121,·)$, $\chi_{504}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{168} a^{6}$, $\frac{1}{168} a^{7}$, $\frac{1}{336} a^{8}$, $\frac{1}{336} a^{9}$, $\frac{1}{672} a^{10}$, $\frac{1}{672} a^{11}$, $\frac{1}{28224} a^{12}$, $\frac{1}{28224} a^{13}$, $\frac{1}{225792} a^{14} + \frac{1}{4}$, $\frac{1}{225792} a^{15} + \frac{1}{4} a$, $\frac{1}{1806336} a^{16} + \frac{1}{903168} a^{14} - \frac{1}{112896} a^{12} - \frac{1}{2688} a^{10} + \frac{1}{1344} a^{8} - \frac{1}{8} a^{4} - \frac{7}{32} a^{2} - \frac{3}{16}$, $\frac{1}{1806336} a^{17} + \frac{1}{903168} a^{15} - \frac{1}{112896} a^{13} - \frac{1}{2688} a^{11} + \frac{1}{1344} a^{9} - \frac{1}{8} a^{5} - \frac{7}{32} a^{3} - \frac{3}{16} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{28}\times C_{364}$, which has order $40768$ (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

\(\Q(\sqrt{-42}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, 3.3.3969.1, \(\Q(\zeta_{7})^+\), 6.0.3456649728.1, 6.0.169375836672.1, 6.0.169375836672.2, 6.0.232339968.1, 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
3Data not computed
7Data not computed