↑

Properties

Label 18.0.15738563230...1696.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{20}\cdot 3^{36}$
Root discriminant $19.44$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $C_2\times C_3:S_4$ (as 18T66)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, 0, 144, 0, 108, 0, -165, 0, -9, 0, 72, 0, -69, 0, 36, 0, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^16 + 36*x^14 - 69*x^12 + 72*x^10 - 9*x^8 - 165*x^6 + 108*x^4 + 144*x^2 + 64)
 
gp: K = bnfinit(x^18 - 9*x^16 + 36*x^14 - 69*x^12 + 72*x^10 - 9*x^8 - 165*x^6 + 108*x^4 + 144*x^2 + 64, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{16} + 36 x^{14} - 69 x^{12} + 72 x^{10} - 9 x^{8} - 165 x^{6} + 108 x^{4} + 144 x^{2} + 64 \)

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:  \(-157385632301186150301696=-\,2^{20}\cdot 3^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.44$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{26} a^{12} - \frac{3}{13} a^{10} + \frac{1}{26} a^{8} + \frac{3}{13} a^{6} - \frac{7}{26} a^{4} - \frac{3}{13} a^{2} - \frac{5}{13}$, $\frac{1}{26} a^{13} - \frac{3}{13} a^{11} + \frac{1}{26} a^{9} + \frac{3}{13} a^{7} - \frac{7}{26} a^{5} - \frac{3}{13} a^{3} - \frac{5}{13} a$, $\frac{1}{52} a^{14} - \frac{1}{52} a^{12} + \frac{5}{26} a^{10} + \frac{11}{52} a^{8} + \frac{5}{26} a^{6} + \frac{11}{52} a^{4} + \frac{25}{52} a^{2} - \frac{6}{13}$, $\frac{1}{208} a^{15} - \frac{1}{104} a^{14} - \frac{1}{208} a^{13} + \frac{1}{104} a^{12} + \frac{9}{52} a^{11} + \frac{2}{13} a^{10} + \frac{11}{208} a^{9} - \frac{11}{104} a^{8} - \frac{1}{13} a^{7} + \frac{2}{13} a^{6} - \frac{41}{208} a^{5} - \frac{11}{104} a^{4} - \frac{53}{208} a^{3} - \frac{51}{104} a^{2} + \frac{7}{52} a - \frac{7}{26}$, $\frac{1}{792896} a^{16} + \frac{1219}{792896} a^{14} - \frac{1}{52} a^{13} + \frac{333}{99112} a^{12} - \frac{7}{52} a^{11} - \frac{113733}{792896} a^{10} + \frac{3}{13} a^{9} - \frac{13341}{198224} a^{8} + \frac{7}{52} a^{7} - \frac{25593}{792896} a^{6} + \frac{5}{13} a^{5} + \frac{73999}{792896} a^{4} - \frac{7}{52} a^{3} + \frac{11211}{24778} a^{2} + \frac{23}{52} a + \frac{21115}{49556}$, $\frac{1}{1585792} a^{17} + \frac{1219}{1585792} a^{15} - \frac{1}{104} a^{14} + \frac{333}{198224} a^{13} + \frac{1}{104} a^{12} - \frac{113733}{1585792} a^{11} + \frac{2}{13} a^{10} - \frac{13341}{396448} a^{9} - \frac{11}{104} a^{8} + \frac{370855}{1585792} a^{7} + \frac{2}{13} a^{6} - \frac{322449}{1585792} a^{5} + \frac{41}{104} a^{4} - \frac{589}{24778} a^{3} + \frac{1}{104} a^{2} + \frac{21115}{99112} a + \frac{3}{13}$

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:  $8$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 64410.9764963 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_4$ (as 18T66):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 18 conjugacy class representatives for $C_2\times C_3:S_4$
Character table for $C_2\times C_3:S_4$

Intermediate fields

3.1.972.2, 3.1.108.1, 3.1.972.1, 3.1.243.1, 6.0.944784.1, 9.1.24794911296.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.16.2$x^{12} - 108 x^{10} - 171 x^{8} + 344 x^{6} - 61 x^{4} + 468 x^{2} + 359$$6$$2$$16$$(C_6\times C_2):C_2$$[2, 2]_{3}^{2}$
3Data not computed