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Properties

Label 18.0.47479756408...1984.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{30}\cdot 7^{15}\cdot 17^{9}$
Root discriminant $206.71$
Ramified primes $2, 3, 7, 17$
Class number $10001880$ (GRH)
Class group $[3, 6, 126, 4410]$ (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![143873586899, 33634384392, 52656339330, 12526883586, 11313506295, 2285310216, 1453703763, 236256312, 121388427, 13600116, 6871230, 360156, 269520, -4380, 7029, -432, 117, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 117*x^16 - 432*x^15 + 7029*x^14 - 4380*x^13 + 269520*x^12 + 360156*x^11 + 6871230*x^10 + 13600116*x^9 + 121388427*x^8 + 236256312*x^7 + 1453703763*x^6 + 2285310216*x^5 + 11313506295*x^4 + 12526883586*x^3 + 52656339330*x^2 + 33634384392*x + 143873586899)
 
gp: K = bnfinit(x^18 - 6*x^17 + 117*x^16 - 432*x^15 + 7029*x^14 - 4380*x^13 + 269520*x^12 + 360156*x^11 + 6871230*x^10 + 13600116*x^9 + 121388427*x^8 + 236256312*x^7 + 1453703763*x^6 + 2285310216*x^5 + 11313506295*x^4 + 12526883586*x^3 + 52656339330*x^2 + 33634384392*x + 143873586899, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 117 x^{16} - 432 x^{15} + 7029 x^{14} - 4380 x^{13} + 269520 x^{12} + 360156 x^{11} + 6871230 x^{10} + 13600116 x^{9} + 121388427 x^{8} + 236256312 x^{7} + 1453703763 x^{6} + 2285310216 x^{5} + 11313506295 x^{4} + 12526883586 x^{3} + 52656339330 x^{2} + 33634384392 x + 143873586899 \)

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:  \(-474797564089947061005643868926528923561984=-\,2^{12}\cdot 3^{30}\cdot 7^{15}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $206.71$
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:  $2, 3, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{10} - \frac{1}{3} a^{8} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{6}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{9} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{8} - \frac{1}{3} a^{6} + \frac{1}{6}$, $\frac{1}{6} a^{13} - \frac{1}{2} a^{8} + \frac{1}{6} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{14} - \frac{1}{3} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{42} a^{15} + \frac{1}{42} a^{14} - \frac{1}{21} a^{13} + \frac{1}{21} a^{12} + \frac{1}{42} a^{11} + \frac{1}{21} a^{10} - \frac{1}{42} a^{9} - \frac{5}{14} a^{8} + \frac{1}{21} a^{7} - \frac{1}{21} a^{6} - \frac{1}{42} a^{5} + \frac{19}{42} a^{4} - \frac{1}{7} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{42} a^{16} - \frac{1}{14} a^{14} - \frac{1}{14} a^{13} - \frac{1}{42} a^{12} + \frac{1}{42} a^{11} - \frac{1}{14} a^{10} + \frac{1}{6} a^{9} + \frac{17}{42} a^{8} + \frac{5}{21} a^{7} + \frac{1}{42} a^{6} + \frac{10}{21} a^{5} + \frac{17}{42} a^{4} - \frac{1}{42} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{1375369053593502538749429230077860719380043639530524137021457494825555954814739782} a^{17} + \frac{2444685622630178400059935525189798227228132870450991829756771231702957205041840}{229228175598917089791571538346310119896673939921754022836909582470925992469123297} a^{16} - \frac{573613557224265544196279843128912771884760483284297560216680353097869866115849}{229228175598917089791571538346310119896673939921754022836909582470925992469123297} a^{15} + \frac{72891101729083434825175959156133472983081070418474645511330404770745797704814063}{1375369053593502538749429230077860719380043639530524137021457494825555954814739782} a^{14} + \frac{819655353548159299310473817136891570832377456104999574375259784284979899691591}{196481293370500362678489890011122959911434805647217733860208213546507993544962826} a^{13} + \frac{89660256379999453612481679131954887803926636191574512059465844356533848057328773}{1375369053593502538749429230077860719380043639530524137021457494825555954814739782} a^{12} - \frac{60357890053276677645205500560188377113697613639477014094648778296092345317041603}{1375369053593502538749429230077860719380043639530524137021457494825555954814739782} a^{11} - \frac{26885066511402659371559382287590462611081823708081791642747732913496483360931869}{687684526796751269374714615038930359690021819765262068510728747412777977407369891} a^{10} + \frac{62293474748121551201694995151863916532236798334645798985570801949648858118099118}{687684526796751269374714615038930359690021819765262068510728747412777977407369891} a^{9} - \frac{114265974231009734187953173096917236012276781618831703835347632915341143733073925}{687684526796751269374714615038930359690021819765262068510728747412777977407369891} a^{8} - \frac{13138268233181748074373220793506116695264317226826510236071543351841734491312389}{65493764456833454226163296670374319970478268549072577953402737848835997848320942} a^{7} - \frac{153522808548982877485494427245647406797938526686984089135745127841191288369042997}{458456351197834179583143076692620239793347879843508045673819164941851984938246594} a^{6} + \frac{24702410413352883560492651069106079624774750928653923281978701166622654828502726}{229228175598917089791571538346310119896673939921754022836909582470925992469123297} a^{5} - \frac{240784713499527413222853577444762481369245673181594801616148353444116323670711247}{687684526796751269374714615038930359690021819765262068510728747412777977407369891} a^{4} + \frac{94488314155220118250424402213687715992180262439149814343916787752726337870446839}{196481293370500362678489890011122959911434805647217733860208213546507993544962826} a^{3} + \frac{10238402888627604266542400431945544497188545765211203327870747401909220335757763}{65493764456833454226163296670374319970478268549072577953402737848835997848320942} a^{2} - \frac{2873780306964211637912936394305061587204716180484996092537663267458383491093540}{98240646685250181339244945005561479955717402823608866930104106773253996772481413} a - \frac{35851406683458134655133033459729778325635496789584552368087803820907998537993286}{98240646685250181339244945005561479955717402823608866930104106773253996772481413}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{126}\times C_{4410}$, which has order $10001880$ (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:  \( 4695974.091249611 \) (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

\(\Q(\sqrt{-119}) \), 3.3.3969.2, 3.3.756.1, 6.0.541760081751.4, 6.0.19655694576.12, 9.9.756284282720064.2

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 ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.9.15.14$x^{9} + 3 x^{7} + 3 x^{6} + 6 x^{3} + 15$$9$$1$$15$$S_3\times C_3$$[3/2, 2]_{2}$
3.9.15.14$x^{9} + 3 x^{7} + 3 x^{6} + 6 x^{3} + 15$$9$$1$$15$$S_3\times C_3$$[3/2, 2]_{2}$
$7$7.6.5.3$x^{6} - 112$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.10.3$x^{12} - 49 x^{6} + 3969$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$17$17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$