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Properties

Label 18.8.43232495562...1152.1
Degree $18$
Signature $[8, 5]$
Discriminant $-\,2^{12}\cdot 3^{27}\cdot 7^{12}$
Root discriminant $30.18$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times A_4^2$ (as 18T109)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27, 0, 0, 0, -189, 0, -36, 0, 378, 0, 126, 0, -180, 0, -63, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 63*x^14 - 180*x^12 + 126*x^10 + 378*x^8 - 36*x^6 - 189*x^4 + 27)
 
gp: K = bnfinit(x^18 - 63*x^14 - 180*x^12 + 126*x^10 + 378*x^8 - 36*x^6 - 189*x^4 + 27, 1)
 

Normalized defining polynomial

\( x^{18} - 63 x^{14} - 180 x^{12} + 126 x^{10} + 378 x^{8} - 36 x^{6} - 189 x^{4} + 27 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-432324955623130532869681152=-\,2^{12}\cdot 3^{27}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.18$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{3} a^{11}$, $\frac{1}{45} a^{12} + \frac{2}{15} a^{6} - \frac{1}{5} a^{4} - \frac{1}{5} a^{2} + \frac{2}{5}$, $\frac{1}{90} a^{13} - \frac{1}{90} a^{12} - \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{10} a^{7} + \frac{1}{10} a^{6} - \frac{1}{10} a^{5} + \frac{1}{10} a^{4} - \frac{1}{10} a^{3} + \frac{1}{10} a^{2} - \frac{3}{10} a + \frac{3}{10}$, $\frac{1}{360} a^{14} + \frac{1}{180} a^{12} - \frac{1}{6} a^{10} - \frac{3}{20} a^{8} - \frac{7}{60} a^{6} + \frac{3}{10} a^{4} - \frac{1}{4} a^{2} + \frac{19}{40}$, $\frac{1}{720} a^{15} - \frac{1}{720} a^{14} + \frac{1}{360} a^{13} - \frac{1}{360} a^{12} - \frac{1}{12} a^{11} + \frac{1}{12} a^{10} + \frac{11}{120} a^{9} - \frac{11}{120} a^{8} + \frac{13}{120} a^{7} - \frac{13}{120} a^{6} - \frac{7}{20} a^{5} + \frac{7}{20} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{19}{80} a - \frac{19}{80}$, $\frac{1}{2880} a^{16} - \frac{1}{960} a^{14} + \frac{1}{288} a^{12} + \frac{61}{480} a^{10} - \frac{1}{240} a^{8} - \frac{9}{160} a^{6} + \frac{5}{32} a^{4} + \frac{149}{320} a^{2} + \frac{13}{64}$, $\frac{1}{5760} a^{17} - \frac{1}{5760} a^{16} - \frac{1}{1920} a^{15} + \frac{1}{1920} a^{14} + \frac{1}{576} a^{13} - \frac{1}{576} a^{12} - \frac{33}{320} a^{11} + \frac{33}{320} a^{10} - \frac{1}{480} a^{9} + \frac{1}{480} a^{8} + \frac{133}{960} a^{7} - \frac{133}{960} a^{6} - \frac{27}{64} a^{5} + \frac{27}{64} a^{4} + \frac{149}{640} a^{3} - \frac{149}{640} a^{2} + \frac{13}{128} a - \frac{13}{128}$

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:  $12$
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:  \( 3104718.63754 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times A_4^2$ (as 18T109):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 288
The 32 conjugacy class representatives for $C_2\times A_4^2$
Character table for $C_2\times A_4^2$ is not computed

Intermediate fields

3.3.3969.2, \(\Q(\zeta_{9})^+\), 3.3.3969.1, \(\Q(\zeta_{7})^+\), 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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
3Data not computed
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$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}$