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Properties

Label 18.6.13380662562...5344.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{24}\cdot 3^{12}\cdot 107^{6}$
Root discriminant $24.88$
Ramified primes $2, 3, 107$
Class number $1$
Class group Trivial
Galois group $C_3:S_3:S_4$ (as 18T154)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, -20, -72, 181, 118, -207, 158, -447, 266, -94, 112, 42, -14, -5, -6, -7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^16 - 6*x^15 - 5*x^14 - 14*x^13 + 42*x^12 + 112*x^11 - 94*x^10 + 266*x^9 - 447*x^8 + 158*x^7 - 207*x^6 + 118*x^5 + 181*x^4 - 72*x^3 - 20*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^18 - 7*x^16 - 6*x^15 - 5*x^14 - 14*x^13 + 42*x^12 + 112*x^11 - 94*x^10 + 266*x^9 - 447*x^8 + 158*x^7 - 207*x^6 + 118*x^5 + 181*x^4 - 72*x^3 - 20*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{16} - 6 x^{15} - 5 x^{14} - 14 x^{13} + 42 x^{12} + 112 x^{11} - 94 x^{10} + 266 x^{9} - 447 x^{8} + 158 x^{7} - 207 x^{6} + 118 x^{5} + 181 x^{4} - 72 x^{3} - 20 x^{2} - 10 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13380662562832253498425344=2^{24}\cdot 3^{12}\cdot 107^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.88$
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:  $2, 3, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{13} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{3} a^{9} + \frac{4}{9} a^{8} + \frac{2}{9} a^{7} + \frac{1}{9} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{4}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{2}{9} a^{9} + \frac{1}{3} a^{7} + \frac{4}{9} a^{6} - \frac{1}{3} a^{5} - \frac{4}{9} a^{4} - \frac{4}{9} a^{3} - \frac{2}{9} a^{2} + \frac{2}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{3} a^{9} - \frac{2}{9} a^{8} + \frac{1}{9} a^{6} + \frac{2}{9} a^{5} + \frac{2}{9} a^{4} - \frac{4}{9} a^{2} + \frac{2}{9} a + \frac{1}{9}$, $\frac{1}{74907351528808671} a^{17} - \frac{1830209881617454}{74907351528808671} a^{16} + \frac{524117350598710}{74907351528808671} a^{15} - \frac{278738797613651}{24969117176269557} a^{14} - \frac{2726707702822312}{24969117176269557} a^{13} + \frac{9187134602987783}{74907351528808671} a^{12} + \frac{9245749880920558}{74907351528808671} a^{11} - \frac{73006623376612}{8323039058756519} a^{10} - \frac{16514921091151333}{74907351528808671} a^{9} + \frac{17619584052203929}{74907351528808671} a^{8} + \frac{29413936037729195}{74907351528808671} a^{7} - \frac{11336662195451189}{74907351528808671} a^{6} - \frac{12349823937011896}{24969117176269557} a^{5} - \frac{6614621622010127}{24969117176269557} a^{4} - \frac{21726664633517455}{74907351528808671} a^{3} - \frac{19209905135273981}{74907351528808671} a^{2} + \frac{10916518110232231}{24969117176269557} a - \frac{7077318938183881}{74907351528808671}$

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:  $11$
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:  \( 598907.304894 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3:S_4$ (as 18T154):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 20 conjugacy class representatives for $C_3:S_3:S_4$
Character table for $C_3:S_3:S_4$

Intermediate fields

3.3.321.1, 6.6.59351616.1, 9.3.6350622912.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.12.9.2$x^{12} - 9 x^{4} + 27$$4$$3$$9$$D_4 \times C_3$$[\ ]_{4}^{6}$
$107$$\Q_{107}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{107}$$x + 3$$1$$1$$0$Trivial$[\ ]$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$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}$