↑

Properties

Label 18.12.6883552242...4224.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,2^{6}\cdot 32009^{6}$
Root discriminant $40.00$
Ramified primes $2, 32009$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times (C_3\times A_4):S_3$ (as 18T156)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32, 288, -280, -1088, 5064, 5204, -13454, -6506, 16119, 1080, -9145, 1960, 2248, -1018, -138, 157, -19, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 - 19*x^16 + 157*x^15 - 138*x^14 - 1018*x^13 + 2248*x^12 + 1960*x^11 - 9145*x^10 + 1080*x^9 + 16119*x^8 - 6506*x^7 - 13454*x^6 + 5204*x^5 + 5064*x^4 - 1088*x^3 - 280*x^2 + 288*x + 32)
 
gp: K = bnfinit(x^18 - 5*x^17 - 19*x^16 + 157*x^15 - 138*x^14 - 1018*x^13 + 2248*x^12 + 1960*x^11 - 9145*x^10 + 1080*x^9 + 16119*x^8 - 6506*x^7 - 13454*x^6 + 5204*x^5 + 5064*x^4 - 1088*x^3 - 280*x^2 + 288*x + 32, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} - 19 x^{16} + 157 x^{15} - 138 x^{14} - 1018 x^{13} + 2248 x^{12} + 1960 x^{11} - 9145 x^{10} + 1080 x^{9} + 16119 x^{8} - 6506 x^{7} - 13454 x^{6} + 5204 x^{5} + 5064 x^{4} - 1088 x^{3} - 280 x^{2} + 288 x + 32 \)

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:  \(-68835522420844686611068124224=-\,2^{6}\cdot 32009^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.00$
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:  $2, 32009$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{40} a^{16} - \frac{3}{40} a^{15} - \frac{1}{40} a^{14} + \frac{3}{40} a^{13} + \frac{1}{10} a^{12} + \frac{1}{20} a^{11} - \frac{3}{10} a^{10} + \frac{1}{10} a^{9} + \frac{3}{8} a^{8} + \frac{3}{20} a^{7} + \frac{11}{40} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{10} a^{3} + \frac{3}{10} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{1273709752818354750400} a^{17} - \frac{10588252592989381199}{1273709752818354750400} a^{16} - \frac{48978274431410838013}{1273709752818354750400} a^{15} - \frac{80177523329632635121}{1273709752818354750400} a^{14} - \frac{10888549032705157729}{79606859551147171900} a^{13} - \frac{60582435140879178501}{636854876409177375200} a^{12} - \frac{55502270106701280841}{318427438204588687600} a^{11} - \frac{26558205028379332689}{79606859551147171900} a^{10} - \frac{164708591392939261689}{1273709752818354750400} a^{9} - \frac{129294636225809430727}{636854876409177375200} a^{8} - \frac{16805826553543882521}{254741950563670950080} a^{7} - \frac{33968443769323040267}{159213719102294343800} a^{6} + \frac{50030499904240433453}{127370975281835475040} a^{5} + \frac{35653145949573577649}{79606859551147171900} a^{4} + \frac{61693838626699825321}{159213719102294343800} a^{3} + \frac{2456297606242613269}{15921371910229434380} a^{2} + \frac{527776524043716461}{31842743820458868760} a + \frac{9442386599474150733}{79606859551147171900}$

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:  \( 194394657.717 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times (C_3\times A_4):S_3$ (as 18T156):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 38 conjugacy class representatives for $C_2\times (C_3\times A_4):S_3$
Character table for $C_2\times (C_3\times A_4):S_3$ is not computed

Intermediate fields

3.3.32009.3, 6.4.4098304324.1, 9.9.32795655776729.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
32009Data not computed