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Properties

Label 18.0.10554808486...1387.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 7^{12}$
Root discriminant $19.01$
Ramified primes $3, 7$
Class number $1$
Class group Trivial
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![1, 0, 0, -11, 0, 0, 125, 0, 0, 42, 0, 0, 27, 0, 0, -4, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^15 + 27*x^12 + 42*x^9 + 125*x^6 - 11*x^3 + 1)
 
gp: K = bnfinit(x^18 - 4*x^15 + 27*x^12 + 42*x^9 + 125*x^6 - 11*x^3 + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{15} + 27 x^{12} + 42 x^{9} + 125 x^{6} - 11 x^{3} + 1 \)

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:  \(-105548084868928352751387=-\,3^{27}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.01$
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:  $3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(63=3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{63}(1,·)$, $\chi_{63}(2,·)$, $\chi_{63}(4,·)$, $\chi_{63}(8,·)$, $\chi_{63}(11,·)$, $\chi_{63}(16,·)$, $\chi_{63}(22,·)$, $\chi_{63}(23,·)$, $\chi_{63}(25,·)$, $\chi_{63}(29,·)$, $\chi_{63}(32,·)$, $\chi_{63}(37,·)$, $\chi_{63}(43,·)$, $\chi_{63}(44,·)$, $\chi_{63}(46,·)$, $\chi_{63}(50,·)$, $\chi_{63}(53,·)$, $\chi_{63}(58,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} + \frac{4}{13} a^{9} - \frac{5}{13} a^{6} + \frac{6}{13} a^{3} - \frac{1}{13}$, $\frac{1}{13} a^{13} + \frac{4}{13} a^{10} - \frac{5}{13} a^{7} + \frac{6}{13} a^{4} - \frac{1}{13} a$, $\frac{1}{13} a^{14} + \frac{4}{13} a^{11} - \frac{5}{13} a^{8} + \frac{6}{13} a^{5} - \frac{1}{13} a^{2}$, $\frac{1}{43303} a^{15} + \frac{826}{43303} a^{12} - \frac{13903}{43303} a^{9} - \frac{4195}{43303} a^{6} + \frac{15825}{43303} a^{3} + \frac{17261}{43303}$, $\frac{1}{43303} a^{16} + \frac{826}{43303} a^{13} - \frac{13903}{43303} a^{10} - \frac{4195}{43303} a^{7} + \frac{15825}{43303} a^{4} + \frac{17261}{43303} a$, $\frac{1}{43303} a^{17} + \frac{826}{43303} a^{14} - \frac{13903}{43303} a^{11} - \frac{4195}{43303} a^{8} + \frac{15825}{43303} a^{5} + \frac{17261}{43303} a^{2}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{524}{43303} a^{17} - \frac{3537}{43303} a^{14} + \frac{19711}{43303} a^{11} - \frac{16375}{43303} a^{8} + \frac{1441}{43303} a^{5} - \frac{218747}{43303} a^{2} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 54408.4888887 \)
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

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{7})^+\), 3.3.3969.1, \(\Q(\zeta_{9})^+\), 3.3.3969.2, 6.0.64827.1, 6.0.47258883.2, \(\Q(\zeta_{9})\), 6.0.47258883.1, 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$