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Properties

Label 18.0.12719225841...3408.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{20}\cdot 3^{9}\cdot 151^{10}$
Root discriminant $60.75$
Ramified primes $2, 3, 151$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![111051, 222021, -92718, -421497, -49059, 344700, 153009, -90513, -97073, -21192, 29125, 14395, 3415, 34, 151, 91, 10, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 10*x^16 + 91*x^15 + 151*x^14 + 34*x^13 + 3415*x^12 + 14395*x^11 + 29125*x^10 - 21192*x^9 - 97073*x^8 - 90513*x^7 + 153009*x^6 + 344700*x^5 - 49059*x^4 - 421497*x^3 - 92718*x^2 + 222021*x + 111051)
 
gp: K = bnfinit(x^18 - 3*x^17 + 10*x^16 + 91*x^15 + 151*x^14 + 34*x^13 + 3415*x^12 + 14395*x^11 + 29125*x^10 - 21192*x^9 - 97073*x^8 - 90513*x^7 + 153009*x^6 + 344700*x^5 - 49059*x^4 - 421497*x^3 - 92718*x^2 + 222021*x + 111051, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 10 x^{16} + 91 x^{15} + 151 x^{14} + 34 x^{13} + 3415 x^{12} + 14395 x^{11} + 29125 x^{10} - 21192 x^{9} - 97073 x^{8} - 90513 x^{7} + 153009 x^{6} + 344700 x^{5} - 49059 x^{4} - 421497 x^{3} - 92718 x^{2} + 222021 x + 111051 \)

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:  \(-127192258415404127890787986833408=-\,2^{20}\cdot 3^{9}\cdot 151^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.75$
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:  $2, 3, 151$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{2}{9} a^{5} + \frac{4}{9} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{4}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{4}$, $\frac{1}{27} a^{11} + \frac{1}{27} a^{10} - \frac{1}{27} a^{9} + \frac{1}{27} a^{8} + \frac{2}{27} a^{7} + \frac{4}{27} a^{6} + \frac{7}{27} a^{5} + \frac{2}{9} a^{4} - \frac{1}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{54} a^{12} - \frac{1}{54} a^{11} - \frac{1}{18} a^{10} - \frac{1}{18} a^{8} + \frac{1}{18} a^{7} - \frac{2}{27} a^{6} - \frac{5}{54} a^{5} - \frac{1}{2} a^{4} - \frac{1}{9} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{54} a^{13} + \frac{1}{54} a^{10} - \frac{1}{54} a^{9} - \frac{1}{27} a^{8} - \frac{5}{54} a^{7} + \frac{1}{54} a^{6} + \frac{1}{27} a^{5} + \frac{7}{18} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{486} a^{14} - \frac{1}{486} a^{13} + \frac{1}{486} a^{12} - \frac{1}{243} a^{11} + \frac{23}{486} a^{10} - \frac{5}{486} a^{9} + \frac{8}{243} a^{8} - \frac{55}{486} a^{7} - \frac{5}{486} a^{6} + \frac{32}{81} a^{5} - \frac{7}{162} a^{4} - \frac{17}{54} a^{3} - \frac{23}{54} a^{2} + \frac{4}{9} a - \frac{1}{3}$, $\frac{1}{486} a^{15} - \frac{1}{486} a^{12} + \frac{1}{162} a^{11} - \frac{25}{486} a^{9} - \frac{1}{162} a^{8} + \frac{2}{81} a^{7} + \frac{7}{486} a^{6} - \frac{1}{54} a^{5} - \frac{38}{81} a^{4} + \frac{1}{27} a^{3} + \frac{19}{54} a^{2} - \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{3402} a^{16} - \frac{1}{3402} a^{14} + \frac{1}{189} a^{13} + \frac{10}{1701} a^{12} - \frac{17}{1701} a^{11} + \frac{1}{567} a^{10} + \frac{82}{1701} a^{9} + \frac{79}{1701} a^{8} + \frac{184}{1701} a^{7} - \frac{11}{1701} a^{6} + \frac{260}{567} a^{5} - \frac{113}{1134} a^{4} + \frac{8}{21} a^{3} - \frac{43}{378} a^{2} - \frac{13}{63} a - \frac{8}{21}$, $\frac{1}{2057732475293455773883803414633042} a^{17} - \frac{23137102812358058374847953889}{158287113484111982606446416510234} a^{16} + \frac{882155892301979579399919600677}{2057732475293455773883803414633042} a^{15} - \frac{227882011721563576461533756939}{342955412548909295647300569105507} a^{14} - \frac{745045005103580216763940836994}{146980891092389698134557386759503} a^{13} - \frac{3793547888862600554933580139370}{1028866237646727886941901707316521} a^{12} + \frac{10821220252504338163745449728829}{685910825097818591294601138211014} a^{11} + \frac{112905109682618225245826369406115}{2057732475293455773883803414633042} a^{10} - \frac{5894439517482440919046221015569}{1028866237646727886941901707316521} a^{9} + \frac{65909485268450530306643448878345}{2057732475293455773883803414633042} a^{8} + \frac{426587826610329562574566291505}{3629157804750362916902651524926} a^{7} - \frac{27295316350371669088685453132782}{342955412548909295647300569105507} a^{6} + \frac{917380378424738035046262048722}{114318470849636431882433523035169} a^{5} + \frac{76670744968788423494418869638}{574464677636363979308711171031} a^{4} + \frac{224206635595685891937704876771}{3629157804750362916902651524926} a^{3} - \frac{5778312336157572059902345961065}{25404104633252540418318560674482} a^{2} - \frac{211386213732866114041843250779}{651387298288526677392783607038} a - \frac{701118893367585622411736885205}{1411339146291807801017697815249}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{297315354830625725}{2182365564981978672063} a^{17} + \frac{1214305770600668876}{2182365564981978672063} a^{16} - \frac{8473984410150345367}{4364731129963957344126} a^{15} - \frac{45362029987017806209}{4364731129963957344126} a^{14} - \frac{39000447423334776965}{4364731129963957344126} a^{13} + \frac{14869738806380916730}{2182365564981978672063} a^{12} - \frac{2055594753431207547001}{4364731129963957344126} a^{11} - \frac{6331028965883345065769}{4364731129963957344126} a^{10} - \frac{5041295140694853050192}{2182365564981978672063} a^{9} + \frac{8210584184673992140823}{1454910376654652448042} a^{8} + \frac{10994171243113242672469}{1454910376654652448042} a^{7} + \frac{800586997084220703217}{242485062775775408007} a^{6} - \frac{12287494082943167480429}{484970125551550816014} a^{5} - \frac{3300969313035205355117}{161656708517183605338} a^{4} + \frac{5287346800635609458101}{161656708517183605338} a^{3} + \frac{669343280074647520630}{26942784752863934223} a^{2} - \frac{481667236537586183171}{26942784752863934223} a - \frac{109867102990607432974}{8980928250954644741} \) (order $6$)
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:  \( 45444000566.5836 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.1812.1, 6.0.9850032.2, 6.0.9850032.3

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

Sibling fields

Degree 12 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.12.16.3$x^{12} - 30 x^{10} - 5 x^{8} + 19 x^{4} + 30 x^{2} + 1$$6$$2$$16$$C_6\times S_3$$[2]_{3}^{6}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$151$151.3.2.2$x^{3} + 755$$3$$1$$2$$C_3$$[\ ]_{3}$
151.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
151.6.3.1$x^{6} - 302 x^{4} + 22801 x^{2} - 86073775$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
151.6.5.3$x^{6} - 94375$$6$$1$$5$$C_6$$[\ ]_{6}$