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Properties

Label 18.6.24496374598...9936.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{18}\cdot 3^{9}\cdot 7^{15}$
Root discriminant $17.53$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
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![1, -8, 15, 17, -63, 39, 15, -21, -54, -54, 117, 45, -23, 22, -3, -5, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 2*x^16 - 5*x^15 - 3*x^14 + 22*x^13 - 23*x^12 + 45*x^11 + 117*x^10 - 54*x^9 - 54*x^8 - 21*x^7 + 15*x^6 + 39*x^5 - 63*x^4 + 17*x^3 + 15*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^18 + 2*x^16 - 5*x^15 - 3*x^14 + 22*x^13 - 23*x^12 + 45*x^11 + 117*x^10 - 54*x^9 - 54*x^8 - 21*x^7 + 15*x^6 + 39*x^5 - 63*x^4 + 17*x^3 + 15*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 2 x^{16} - 5 x^{15} - 3 x^{14} + 22 x^{13} - 23 x^{12} + 45 x^{11} + 117 x^{10} - 54 x^{9} - 54 x^{8} - 21 x^{7} + 15 x^{6} + 39 x^{5} - 63 x^{4} + 17 x^{3} + 15 x^{2} - 8 x + 1 \)

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:  \(24496374598915344039936=2^{18}\cdot 3^{9}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.53$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{15} a^{15} + \frac{1}{15} a^{12} + \frac{2}{15} a^{11} - \frac{1}{15} a^{9} + \frac{2}{15} a^{8} - \frac{1}{15} a^{7} - \frac{4}{15} a^{6} + \frac{1}{3} a^{5} + \frac{2}{5} a^{4} + \frac{1}{15} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{15}$, $\frac{1}{45} a^{16} + \frac{1}{45} a^{15} + \frac{1}{9} a^{14} - \frac{4}{45} a^{13} + \frac{1}{15} a^{12} + \frac{2}{45} a^{11} - \frac{2}{15} a^{10} + \frac{1}{45} a^{9} + \frac{1}{45} a^{8} + \frac{1}{3} a^{7} - \frac{4}{45} a^{6} + \frac{11}{45} a^{5} - \frac{2}{5} a^{4} - \frac{17}{45} a^{3} - \frac{2}{45} a^{2} + \frac{4}{9} a + \frac{4}{45}$, $\frac{1}{2115940095} a^{17} - \frac{5925712}{2115940095} a^{16} + \frac{4092269}{235104455} a^{15} + \frac{4842356}{51608295} a^{14} - \frac{47617868}{423188019} a^{13} - \frac{44599873}{2115940095} a^{12} + \frac{288638171}{2115940095} a^{11} - \frac{70675631}{2115940095} a^{10} + \frac{348914594}{2115940095} a^{9} + \frac{10156910}{423188019} a^{8} - \frac{397719853}{2115940095} a^{7} + \frac{294451567}{2115940095} a^{6} + \frac{193197044}{2115940095} a^{5} + \frac{757031191}{2115940095} a^{4} - \frac{501480082}{2115940095} a^{3} + \frac{137104358}{705313365} a^{2} - \frac{19550264}{705313365} a - \frac{960084101}{2115940095}$

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:  \( 14420.859848954282 \)
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

\(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), 3.1.1176.1, 6.2.29042496.1, \(\Q(\zeta_{21})^+\), 9.3.1626379776.1

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$$[\ ]^{6}$
2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed