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Properties

Label 18.12.6042409957...8352.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,2^{6}\cdot 3^{33}\cdot 19^{8}$
Root discriminant $34.95$
Ramified primes $2, 3, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T585

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 0, -45, 0, 216, 0, -396, 0, 153, 0, 207, 0, -51, 0, -27, 0, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 3*x^16 - 27*x^14 - 51*x^12 + 207*x^10 + 153*x^8 - 396*x^6 + 216*x^4 - 45*x^2 + 3)
 
gp: K = bnfinit(x^18 + 3*x^16 - 27*x^14 - 51*x^12 + 207*x^10 + 153*x^8 - 396*x^6 + 216*x^4 - 45*x^2 + 3, 1)
 

Normalized defining polynomial

\( x^{18} + 3 x^{16} - 27 x^{14} - 51 x^{12} + 207 x^{10} + 153 x^{8} - 396 x^{6} + 216 x^{4} - 45 x^{2} + 3 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-6042409957173305670222428352=-\,2^{6}\cdot 3^{33}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.95$
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:  $2, 3, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{20} a^{12} - \frac{1}{5} a^{10} - \frac{1}{4} a^{9} + \frac{1}{5} a^{8} - \frac{1}{4} a^{7} - \frac{9}{20} a^{6} + \frac{1}{4} a^{5} - \frac{1}{20} a^{4} - \frac{1}{2} a^{3} - \frac{9}{20} a^{2} + \frac{1}{4} a - \frac{1}{5}$, $\frac{1}{20} a^{13} + \frac{1}{20} a^{11} - \frac{1}{4} a^{10} + \frac{1}{5} a^{9} + \frac{1}{20} a^{7} - \frac{3}{10} a^{5} + \frac{1}{4} a^{4} - \frac{1}{5} a^{3} + \frac{1}{4} a^{2} + \frac{1}{20} a + \frac{1}{4}$, $\frac{1}{20} a^{14} - \frac{1}{10} a^{10} - \frac{1}{4} a^{9} + \frac{1}{10} a^{8} - \frac{1}{4} a^{7} + \frac{2}{5} a^{6} + \frac{1}{4} a^{5} + \frac{1}{10} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{9}{20}$, $\frac{1}{40} a^{15} - \frac{1}{40} a^{14} - \frac{1}{40} a^{13} - \frac{1}{40} a^{12} + \frac{1}{20} a^{11} - \frac{1}{10} a^{10} + \frac{1}{5} a^{9} - \frac{3}{20} a^{8} + \frac{7}{40} a^{7} + \frac{1}{40} a^{6} + \frac{13}{40} a^{5} - \frac{11}{40} a^{4} - \frac{1}{40} a^{3} - \frac{11}{40} a^{2} - \frac{17}{40} a + \frac{1}{8}$, $\frac{1}{80} a^{16} + \frac{1}{80} a^{12} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{16} a^{8} - \frac{1}{4} a^{7} + \frac{3}{8} a^{6} + \frac{1}{4} a^{5} + \frac{1}{40} a^{4} - \frac{1}{2} a^{3} + \frac{1}{40} a^{2} + \frac{1}{4} a - \frac{7}{80}$, $\frac{1}{160} a^{17} - \frac{1}{160} a^{16} - \frac{3}{160} a^{13} + \frac{3}{160} a^{12} + \frac{3}{80} a^{11} - \frac{3}{80} a^{10} + \frac{29}{160} a^{9} + \frac{11}{160} a^{8} + \frac{13}{80} a^{7} - \frac{33}{80} a^{6} - \frac{17}{80} a^{5} - \frac{3}{80} a^{4} + \frac{19}{80} a^{3} - \frac{19}{80} a^{2} - \frac{31}{160} a + \frac{71}{160}$

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:  $14$
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:  \( 58226863.3829 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T585:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 13824
The 96 conjugacy class representatives for t18n585 are not computed
Character table for t18n585 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.5609891727441.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 siblings: data not computed

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
3Data not computed
$19$19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.6.4.2$x^{6} - 19 x^{3} + 722$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.6.4.1$x^{6} + 57 x^{3} + 1444$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$