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Properties

Label 18.2.11431756166...5017.1
Degree $18$
Signature $[2, 8]$
Discriminant $3^{6}\cdot 97^{3}\cdot 107^{8}$
Root discriminant $24.67$
Ramified primes $3, 97, 107$
Class number $1$
Class group Trivial
Galois group $C_2\times S_3\times S_4$ (as 18T111)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -108, 646, -1269, 2108, -2775, 2800, -2403, 1709, -1168, 978, -800, 556, -319, 132, -48, 15, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 15*x^16 - 48*x^15 + 132*x^14 - 319*x^13 + 556*x^12 - 800*x^11 + 978*x^10 - 1168*x^9 + 1709*x^8 - 2403*x^7 + 2800*x^6 - 2775*x^5 + 2108*x^4 - 1269*x^3 + 646*x^2 - 108*x + 8)
 
gp: K = bnfinit(x^18 - 4*x^17 + 15*x^16 - 48*x^15 + 132*x^14 - 319*x^13 + 556*x^12 - 800*x^11 + 978*x^10 - 1168*x^9 + 1709*x^8 - 2403*x^7 + 2800*x^6 - 2775*x^5 + 2108*x^4 - 1269*x^3 + 646*x^2 - 108*x + 8, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 15 x^{16} - 48 x^{15} + 132 x^{14} - 319 x^{13} + 556 x^{12} - 800 x^{11} + 978 x^{10} - 1168 x^{9} + 1709 x^{8} - 2403 x^{7} + 2800 x^{6} - 2775 x^{5} + 2108 x^{4} - 1269 x^{3} + 646 x^{2} - 108 x + 8 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11431756166378830117885017=3^{6}\cdot 97^{3}\cdot 107^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.67$
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:  $3, 97, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{7} a^{11} + \frac{2}{7} a^{10} - \frac{1}{7} a^{9} + \frac{1}{7} a^{8} - \frac{3}{7} a^{6} + \frac{2}{7} a^{5} - \frac{2}{7} a^{4} + \frac{2}{7} a^{3} - \frac{3}{7} a^{2} + \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{12} + \frac{2}{7} a^{10} + \frac{3}{7} a^{9} - \frac{2}{7} a^{8} - \frac{3}{7} a^{7} + \frac{1}{7} a^{6} + \frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{14} a^{13} - \frac{1}{14} a^{10} + \frac{1}{7} a^{8} + \frac{1}{14} a^{7} - \frac{5}{14} a^{5} - \frac{3}{14} a^{4} - \frac{2}{7} a^{3} + \frac{1}{14} a^{2} - \frac{3}{14} a + \frac{3}{7}$, $\frac{1}{42} a^{14} + \frac{1}{42} a^{13} + \frac{1}{21} a^{12} - \frac{1}{14} a^{11} + \frac{13}{42} a^{10} + \frac{5}{21} a^{9} - \frac{17}{42} a^{8} - \frac{19}{42} a^{7} + \frac{17}{42} a^{6} + \frac{2}{21} a^{5} - \frac{5}{42} a^{4} + \frac{1}{6} a^{3} + \frac{3}{7} a^{2} + \frac{19}{42} a - \frac{2}{21}$, $\frac{1}{42} a^{15} + \frac{1}{42} a^{13} + \frac{1}{42} a^{12} - \frac{1}{21} a^{11} + \frac{5}{14} a^{10} + \frac{3}{14} a^{9} + \frac{5}{21} a^{8} + \frac{3}{7} a^{7} + \frac{5}{42} a^{6} + \frac{1}{14} a^{5} + \frac{17}{42} a^{3} + \frac{13}{42} a^{2} + \frac{13}{42} a + \frac{5}{21}$, $\frac{1}{7056} a^{16} - \frac{1}{294} a^{15} + \frac{83}{7056} a^{14} + \frac{29}{1764} a^{13} - \frac{2}{147} a^{12} - \frac{109}{2352} a^{11} - \frac{7}{18} a^{10} + \frac{47}{252} a^{9} - \frac{1051}{3528} a^{8} + \frac{83}{882} a^{7} + \frac{677}{7056} a^{6} - \frac{2735}{7056} a^{5} + \frac{10}{21} a^{4} + \frac{2981}{7056} a^{3} + \frac{4}{9} a^{2} - \frac{2929}{7056} a - \frac{767}{3528}$, $\frac{1}{20825380786629888} a^{17} - \frac{599294507437}{10412690393314944} a^{16} + \frac{198269790001739}{20825380786629888} a^{15} - \frac{6384152227207}{1487527199044992} a^{14} - \frac{10235440638001}{1301586299164368} a^{13} + \frac{281425181105419}{6941793595543296} a^{12} + \frac{537485731071755}{10412690393314944} a^{11} + \frac{43205618509379}{247921199840832} a^{10} - \frac{1379217810796537}{3470896797771648} a^{9} - \frac{814579306316395}{5206345196657472} a^{8} + \frac{778158970693543}{6941793595543296} a^{7} + \frac{103397981072069}{6941793595543296} a^{6} + \frac{5184817530185843}{10412690393314944} a^{5} - \frac{1913972566048411}{20825380786629888} a^{4} + \frac{57737978373935}{1156965599257216} a^{3} - \frac{2748442179747811}{6941793595543296} a^{2} + \frac{344559584184307}{5206345196657472} a + \frac{1808765451508931}{5206345196657472}$

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:  $9$
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:  \( 178327.014477 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3\times S_4$ (as 18T111):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 288
The 30 conjugacy class representatives for $C_2\times S_3\times S_4$
Character table for $C_2\times S_3\times S_4$ is not computed

Intermediate fields

3.1.107.1, 3.3.321.1, 6.2.9994977.1, 9.3.3539149227.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
97Data not computed
$107$107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$