Properties

Label 18.18.3181842192...4336.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{18}\cdot 3^{24}\cdot 73^{10}$
Root discriminant $93.84$
Ramified primes $2, 3, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_6^2:C_3$ (as 18T48)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5329, 0, 31974, 0, -79935, 0, 107675, 0, -84315, 0, 38544, 0, -9772, 0, 1221, 0, -63, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 63*x^16 + 1221*x^14 - 9772*x^12 + 38544*x^10 - 84315*x^8 + 107675*x^6 - 79935*x^4 + 31974*x^2 - 5329)
 
gp: K = bnfinit(x^18 - 63*x^16 + 1221*x^14 - 9772*x^12 + 38544*x^10 - 84315*x^8 + 107675*x^6 - 79935*x^4 + 31974*x^2 - 5329, 1)
 

Normalized defining polynomial

\( x^{18} - 63 x^{16} + 1221 x^{14} - 9772 x^{12} + 38544 x^{10} - 84315 x^{8} + 107675 x^{6} - 79935 x^{4} + 31974 x^{2} - 5329 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(318184219227442073683553705831694336=2^{18}\cdot 3^{24}\cdot 73^{10}\)
magma: Discriminant(K);
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.84$
magma: Abs(Discriminant(K))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 73$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{292} a^{12} + \frac{5}{146} a^{10} + \frac{53}{292} a^{8} + \frac{5}{146} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4}$, $\frac{1}{292} a^{13} + \frac{5}{146} a^{11} + \frac{53}{292} a^{9} + \frac{5}{146} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a$, $\frac{1}{292} a^{14} - \frac{47}{292} a^{10} + \frac{16}{73} a^{8} + \frac{23}{146} a^{6} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{292} a^{15} - \frac{47}{292} a^{11} + \frac{16}{73} a^{9} + \frac{23}{146} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{52268} a^{16} - \frac{15}{13067} a^{14} - \frac{33}{52268} a^{12} - \frac{2602}{13067} a^{10} + \frac{40}{13067} a^{8} + \frac{953}{26134} a^{6} + \frac{87}{716} a^{4} - \frac{59}{358} a^{2} + \frac{21}{179}$, $\frac{1}{52268} a^{17} - \frac{15}{13067} a^{15} - \frac{33}{52268} a^{13} - \frac{2602}{13067} a^{11} + \frac{40}{13067} a^{9} + \frac{953}{26134} a^{7} + \frac{87}{716} a^{5} - \frac{59}{358} a^{3} + \frac{21}{179} a$

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:  $17$
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:  \( 1494891798290 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6^2:C_3$ (as 18T48):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_6^2:C_3$
Character table for $C_6^2:C_3$

Intermediate fields

\(\Q(\zeta_{9})^+\), 6.6.2237668416.1, 9.9.15091989595281.1

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$73$73.6.5.3$x^{6} - 45625$$6$$1$$5$$C_6$$[\ ]_{6}$
73.6.5.6$x^{6} + 228125$$6$$1$$5$$C_6$$[\ ]_{6}$
73.6.0.1$x^{6} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$