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Properties

Label 18.0.60940886311...3424.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{33}\cdot 3^{21}\cdot 7^{14}$
Root discriminant $58.32$
Ramified primes $2, 3, 7$
Class number $1134$ (GRH)
Class group $[3, 3, 126]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![216, 0, 8208, 0, 52272, 0, 96516, 0, 77112, 0, 31500, 0, 6972, 0, 828, 0, 48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 48*x^16 + 828*x^14 + 6972*x^12 + 31500*x^10 + 77112*x^8 + 96516*x^6 + 52272*x^4 + 8208*x^2 + 216)
 
gp: K = bnfinit(x^18 + 48*x^16 + 828*x^14 + 6972*x^12 + 31500*x^10 + 77112*x^8 + 96516*x^6 + 52272*x^4 + 8208*x^2 + 216, 1)
 

Normalized defining polynomial

\( x^{18} + 48 x^{16} + 828 x^{14} + 6972 x^{12} + 31500 x^{10} + 77112 x^{8} + 96516 x^{6} + 52272 x^{4} + 8208 x^{2} + 216 \)

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:  \(-60940886311952006474852072423424=-\,2^{33}\cdot 3^{21}\cdot 7^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.32$
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 a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6} a^{6}$, $\frac{1}{6} a^{7}$, $\frac{1}{6} a^{8}$, $\frac{1}{6} a^{9}$, $\frac{1}{6} a^{10}$, $\frac{1}{6} a^{11}$, $\frac{1}{252} a^{12} + \frac{1}{14} a^{10} - \frac{1}{21} a^{8} - \frac{1}{42} a^{6} - \frac{3}{7} a^{4} + \frac{2}{7} a^{2} + \frac{1}{7}$, $\frac{1}{252} a^{13} + \frac{1}{14} a^{11} - \frac{1}{21} a^{9} - \frac{1}{42} a^{7} - \frac{3}{7} a^{5} + \frac{2}{7} a^{3} + \frac{1}{7} a$, $\frac{1}{252} a^{14} + \frac{3}{7}$, $\frac{1}{252} a^{15} + \frac{3}{7} a$, $\frac{1}{994695156} a^{16} - \frac{294865}{497347578} a^{14} - \frac{489695}{248673789} a^{12} - \frac{788311}{12752502} a^{10} + \frac{1741471}{27630421} a^{8} + \frac{104711}{165782526} a^{6} + \frac{8179417}{27630421} a^{4} - \frac{225193}{3947203} a^{2} - \frac{3371096}{27630421}$, $\frac{1}{994695156} a^{17} - \frac{294865}{497347578} a^{15} - \frac{489695}{248673789} a^{13} - \frac{788311}{12752502} a^{11} + \frac{1741471}{27630421} a^{9} + \frac{104711}{165782526} a^{7} + \frac{8179417}{27630421} a^{5} - \frac{225193}{3947203} a^{3} - \frac{3371096}{27630421} a$

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

Class group and class number

$C_{3}\times C_{3}\times C_{126}$, which has order $1134$ (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:  \( 296124.35954857944 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-6}) \), \(\Q(\zeta_{7})^+\), 3.3.756.1, 6.0.33191424.1, 6.0.219469824.1, 9.9.1037426999616.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 sibling: 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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
2Data not computed
3Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$