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Properties

Label 18.2.81879229526...6288.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{30}\cdot 3^{27}$
Root discriminant $16.50$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group 18T544

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-27, 0, 81, 0, 135, 0, -63, 0, -81, 0, 9, 0, 6, 0, 0, 0, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 3*x^16 + 6*x^12 + 9*x^10 - 81*x^8 - 63*x^6 + 135*x^4 + 81*x^2 - 27)
 
gp: K = bnfinit(x^18 + 3*x^16 + 6*x^12 + 9*x^10 - 81*x^8 - 63*x^6 + 135*x^4 + 81*x^2 - 27, 1)
 

Normalized defining polynomial

\( x^{18} + 3 x^{16} + 6 x^{12} + 9 x^{10} - 81 x^{8} - 63 x^{6} + 135 x^{4} + 81 x^{2} - 27 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8187922952619753996288=2^{30}\cdot 3^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.50$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{3} a^{11}$, $\frac{1}{9} a^{12}$, $\frac{1}{18} a^{13} - \frac{1}{18} a^{12} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{36} a^{14} + \frac{1}{36} a^{12} - \frac{1}{12} a^{10} + \frac{1}{12} a^{8} - \frac{1}{6} a^{6} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{72} a^{15} - \frac{1}{72} a^{14} + \frac{1}{72} a^{13} - \frac{1}{72} a^{12} + \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} + \frac{1}{12} a^{7} - \frac{1}{12} a^{6} + \frac{3}{8} a^{3} - \frac{3}{8} a^{2} + \frac{1}{8} a - \frac{1}{8}$, $\frac{1}{23472} a^{16} + \frac{67}{5868} a^{14} + \frac{151}{5868} a^{12} + \frac{1}{24} a^{10} + \frac{329}{7824} a^{8} + \frac{547}{3912} a^{6} + \frac{1003}{2608} a^{4} - \frac{103}{1304} a^{2} + \frac{187}{2608}$, $\frac{1}{46944} a^{17} - \frac{1}{46944} a^{16} + \frac{67}{11736} a^{15} - \frac{67}{11736} a^{14} + \frac{151}{11736} a^{13} - \frac{151}{11736} a^{12} + \frac{1}{48} a^{11} - \frac{1}{48} a^{10} + \frac{329}{15648} a^{9} - \frac{329}{15648} a^{8} + \frac{547}{7824} a^{7} - \frac{547}{7824} a^{6} + \frac{1003}{5216} a^{5} - \frac{1003}{5216} a^{4} - \frac{103}{2608} a^{3} + \frac{103}{2608} a^{2} + \frac{187}{5216} a - \frac{187}{5216}$

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:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4683.37860456 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T544:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 9216
The 96 conjugacy class representatives for t18n544 are not computed
Character table for t18n544 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.1.648.1, 9.3.272097792.1

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 ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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}$
2.12.30.112$x^{12} - 6 x^{10} + x^{8} + 4 x^{6} + 3 x^{4} + 2 x^{2} + 3$$4$$3$$30$12T134$[2, 2, 2, 3, 7/2, 7/2, 7/2]^{3}$
3Data not computed