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Properties

Label 18.18.2443628984...8016.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{24}\cdot 3^{20}\cdot 11^{15}$
Root discriminant $63.00$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\wr C_2$ (as 18T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1756, -2688, 44124, 67824, -145320, -190776, 156600, 196728, -76188, -95540, 20004, 23940, -3519, -3228, 441, 222, -33, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 33*x^16 + 222*x^15 + 441*x^14 - 3228*x^13 - 3519*x^12 + 23940*x^11 + 20004*x^10 - 95540*x^9 - 76188*x^8 + 196728*x^7 + 156600*x^6 - 190776*x^5 - 145320*x^4 + 67824*x^3 + 44124*x^2 - 2688*x - 1756)
 
gp: K = bnfinit(x^18 - 6*x^17 - 33*x^16 + 222*x^15 + 441*x^14 - 3228*x^13 - 3519*x^12 + 23940*x^11 + 20004*x^10 - 95540*x^9 - 76188*x^8 + 196728*x^7 + 156600*x^6 - 190776*x^5 - 145320*x^4 + 67824*x^3 + 44124*x^2 - 2688*x - 1756, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 33 x^{16} + 222 x^{15} + 441 x^{14} - 3228 x^{13} - 3519 x^{12} + 23940 x^{11} + 20004 x^{10} - 95540 x^{9} - 76188 x^{8} + 196728 x^{7} + 156600 x^{6} - 190776 x^{5} - 145320 x^{4} + 67824 x^{3} + 44124 x^{2} - 2688 x - 1756 \)

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:  \(244362898413546378439384960598016=2^{24}\cdot 3^{20}\cdot 11^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.00$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{13} + \frac{1}{4} a^{11} - \frac{1}{2} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} + \frac{3}{8} a^{10} + \frac{1}{8} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} + \frac{1}{4} a^{10} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{40} a^{16} + \frac{1}{40} a^{15} + \frac{3}{40} a^{13} + \frac{1}{20} a^{12} + \frac{19}{40} a^{11} - \frac{7}{20} a^{10} + \frac{11}{40} a^{9} + \frac{7}{40} a^{8} - \frac{1}{2} a^{7} - \frac{2}{5} a^{6} + \frac{3}{20} a^{5} + \frac{3}{20} a^{4} - \frac{9}{20} a^{3} + \frac{1}{4} a^{2} - \frac{1}{10} a + \frac{2}{5}$, $\frac{1}{607317623783646194514622840} a^{17} + \frac{6170590458779825036923491}{607317623783646194514622840} a^{16} - \frac{1480536997240560437873639}{60731762378364619451462284} a^{15} + \frac{8844362524539624979436177}{151829405945911548628655710} a^{14} - \frac{12087004509452390224154889}{303658811891823097257311420} a^{13} - \frac{6214993139316432998612853}{303658811891823097257311420} a^{12} + \frac{75235934857885998854617519}{151829405945911548628655710} a^{11} - \frac{150961801443612549055183957}{303658811891823097257311420} a^{10} - \frac{301680851224250927521846933}{607317623783646194514622840} a^{9} + \frac{13509294519472920099766311}{121463524756729238902924568} a^{8} - \frac{134612223763977422981428393}{303658811891823097257311420} a^{7} - \frac{69737275436926887622725357}{303658811891823097257311420} a^{6} - \frac{29310707239270546280985563}{75914702972955774314327855} a^{5} + \frac{3956647676326966152526651}{303658811891823097257311420} a^{4} - \frac{29679812570876496385791315}{60731762378364619451462284} a^{3} + \frac{122831718143925359774260353}{303658811891823097257311420} a^{2} - \frac{30837885825456277639363368}{75914702972955774314327855} a - \frac{13757966602221702717521347}{30365881189182309725731142}$

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:  \( 126944040549 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\wr C_2$ (as 18T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $S_3\wr C_2$
Character table for $S_3\wr C_2$

Intermediate fields

\(\Q(\sqrt{11}) \), 9.9.1178314711428096.1

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

Sibling fields

Degree 6 siblings: data not computed
Degree 9 sibling: data not computed
Degree 12 siblings: data not computed
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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.8.3$x^{6} + 2 x^{3} + 6$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.12.16.13$x^{12} + 12 x^{10} + 12 x^{8} + 8 x^{6} + 32 x^{4} - 16 x^{2} + 16$$6$$2$$16$$D_6$$[2]_{3}^{2}$
3Data not computed
$11$11.6.5.1$x^{6} - 11$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
11.12.10.1$x^{12} + 3146 x^{6} + 14235529$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$