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Properties

Label 18.6.52310459739...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 5^{8}\cdot 83^{6}$
Root discriminant $14.16$
Ramified primes $2, 5, 83$
Class number $1$
Class group Trivial
Galois group $(C_3\times A_4):S_3$ (as 18T108)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, -3, 0, -7, 0, 18, 0, -7, 0, 2, 0, 5, 0, -6, 0, -1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^16 - 6*x^14 + 5*x^12 + 2*x^10 - 7*x^8 + 18*x^6 - 7*x^4 - 3*x^2 - 1)
 
gp: K = bnfinit(x^18 - x^16 - 6*x^14 + 5*x^12 + 2*x^10 - 7*x^8 + 18*x^6 - 7*x^4 - 3*x^2 - 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{16} - 6 x^{14} + 5 x^{12} + 2 x^{10} - 7 x^{8} + 18 x^{6} - 7 x^{4} - 3 x^{2} - 1 \)

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:  \(523104597390400000000=2^{12}\cdot 5^{8}\cdot 83^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.16$
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:  $2, 5, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{14} - \frac{1}{6} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{582} a^{16} - \frac{7}{194} a^{14} - \frac{1}{6} a^{13} - \frac{71}{582} a^{12} - \frac{1}{3} a^{11} + \frac{82}{291} a^{10} + \frac{1}{3} a^{9} + \frac{10}{291} a^{8} - \frac{1}{3} a^{7} - \frac{71}{194} a^{6} - \frac{1}{3} a^{5} - \frac{92}{291} a^{4} + \frac{1}{6} a^{3} + \frac{14}{97} a^{2} - \frac{1}{6} a - \frac{131}{582}$, $\frac{1}{582} a^{17} - \frac{7}{194} a^{15} - \frac{1}{6} a^{14} - \frac{71}{582} a^{13} + \frac{82}{291} a^{11} + \frac{10}{291} a^{9} - \frac{71}{194} a^{7} + \frac{1}{3} a^{6} - \frac{92}{291} a^{5} - \frac{1}{6} a^{4} + \frac{14}{97} a^{3} - \frac{1}{2} a^{2} - \frac{131}{582} a + \frac{1}{3}$

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:  \( 1833.2983103 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_3\times A_4):S_3$ (as 18T108):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 216
The 19 conjugacy class representatives for $(C_3\times A_4):S_3$
Character table for $(C_3\times A_4):S_3$

Intermediate fields

3.1.83.1, 6.2.110224.1, 9.3.357366875.1

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

Sibling fields

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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
$5$5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.12.8.2$x^{12} + 25 x^{6} - 250 x^{3} + 1250$$3$$4$$8$$C_3\times (C_3 : C_4)$$[\ ]_{3}^{12}$
$83$83.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
83.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
83.12.6.1$x^{12} + 38881516 x^{6} - 3939040643 x^{2} + 377943071614564$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$