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Properties

Label 18.0.13509211106...4707.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 11^{6}$
Root discriminant $11.56$
Ramified primes $3, 11$
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, -6, 15, -11, -33, 84, -18, -138, 108, 126, -162, -51, 131, -21, -42, 14, 9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 9*x^16 + 14*x^15 - 42*x^14 - 21*x^13 + 131*x^12 - 51*x^11 - 162*x^10 + 126*x^9 + 108*x^8 - 138*x^7 - 18*x^6 + 84*x^5 - 33*x^4 - 11*x^3 + 15*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 9*x^16 + 14*x^15 - 42*x^14 - 21*x^13 + 131*x^12 - 51*x^11 - 162*x^10 + 126*x^9 + 108*x^8 - 138*x^7 - 18*x^6 + 84*x^5 - 33*x^4 - 11*x^3 + 15*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 9 x^{16} + 14 x^{15} - 42 x^{14} - 21 x^{13} + 131 x^{12} - 51 x^{11} - 162 x^{10} + 126 x^{9} + 108 x^{8} - 138 x^{7} - 18 x^{6} + 84 x^{5} - 33 x^{4} - 11 x^{3} + 15 x^{2} - 6 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-13509211106101054707=-\,3^{27}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.56$
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:  $3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2464735157} a^{17} + \frac{1187851471}{2464735157} a^{16} - \frac{1093584224}{2464735157} a^{15} + \frac{45528195}{144984421} a^{14} - \frac{359164465}{2464735157} a^{13} - \frac{856259767}{2464735157} a^{12} - \frac{33364627}{129722903} a^{11} + \frac{582676255}{2464735157} a^{10} - \frac{571679900}{2464735157} a^{9} + \frac{604063709}{2464735157} a^{8} + \frac{1110616377}{2464735157} a^{7} - \frac{299591893}{2464735157} a^{6} - \frac{331648201}{2464735157} a^{5} - \frac{973264069}{2464735157} a^{4} + \frac{15336687}{129722903} a^{3} - \frac{324987981}{2464735157} a^{2} - \frac{81290624}{2464735157} a + \frac{258328831}{2464735157}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{981046}{129722903} a^{17} + \frac{96120271}{129722903} a^{16} - \frac{495557418}{129722903} a^{15} + \frac{28308669}{7630759} a^{14} + \frac{1696481148}{129722903} a^{13} - \frac{2762676003}{129722903} a^{12} - \frac{4190165281}{129722903} a^{11} + \frac{9664567384}{129722903} a^{10} + \frac{2745080357}{129722903} a^{9} - \frac{14379420820}{129722903} a^{8} + \frac{517675004}{129722903} a^{7} + \frac{12700331666}{129722903} a^{6} - \frac{3266221525}{129722903} a^{5} - \frac{5881514363}{129722903} a^{4} + \frac{3453120424}{129722903} a^{3} + \frac{526453749}{129722903} a^{2} - \frac{1075070733}{129722903} a + \frac{423662918}{129722903} \) (order $18$)
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:  \( 835.1622079049273 \)
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

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), 3.1.891.1, \(\Q(\zeta_{9})\), 6.0.2381643.1, 9.3.707347971.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$11$11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$