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Properties

Label 18.0.64767749918...9472.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{20}\cdot 3^{31}$
Root discriminant $14.33$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 39, -108, 228, -420, 696, -978, 1143, -1183, 1143, -978, 696, -420, 228, -108, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 228*x^14 - 420*x^13 + 696*x^12 - 978*x^11 + 1143*x^10 - 1183*x^9 + 1143*x^8 - 978*x^7 + 696*x^6 - 420*x^5 + 228*x^4 - 108*x^3 + 39*x^2 - 9*x + 1)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 228*x^14 - 420*x^13 + 696*x^12 - 978*x^11 + 1143*x^10 - 1183*x^9 + 1143*x^8 - 978*x^7 + 696*x^6 - 420*x^5 + 228*x^4 - 108*x^3 + 39*x^2 - 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 108 x^{15} + 228 x^{14} - 420 x^{13} + 696 x^{12} - 978 x^{11} + 1143 x^{10} - 1183 x^{9} + 1143 x^{8} - 978 x^{7} + 696 x^{6} - 420 x^{5} + 228 x^{4} - 108 x^{3} + 39 x^{2} - 9 x + 1 \)

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:  \(-647677499181836009472=-\,2^{20}\cdot 3^{31}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.33$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{106} a^{15} + \frac{19}{106} a^{14} + \frac{8}{53} a^{13} + \frac{23}{106} a^{12} + \frac{29}{106} a^{11} - \frac{20}{53} a^{10} + \frac{47}{106} a^{9} + \frac{37}{106} a^{8} + \frac{37}{106} a^{7} - \frac{3}{53} a^{6} + \frac{13}{106} a^{5} + \frac{29}{106} a^{4} - \frac{15}{53} a^{3} - \frac{37}{106} a^{2} + \frac{19}{106} a - \frac{26}{53}$, $\frac{1}{212} a^{16} + \frac{13}{106} a^{14} - \frac{4}{53} a^{13} + \frac{4}{53} a^{12} - \frac{2}{53} a^{11} - \frac{47}{106} a^{10} + \frac{49}{106} a^{9} + \frac{23}{212} a^{8} - \frac{5}{53} a^{7} + \frac{37}{106} a^{6} - \frac{3}{106} a^{5} - \frac{26}{53} a^{4} + \frac{14}{53} a^{3} + \frac{43}{106} a^{2} - \frac{21}{106} a - \frac{19}{212}$, $\frac{1}{212} a^{17} + \frac{5}{53} a^{14} + \frac{6}{53} a^{13} + \frac{15}{106} a^{12} - \frac{7}{53} a^{10} + \frac{73}{212} a^{9} - \frac{7}{53} a^{8} - \frac{10}{53} a^{7} + \frac{11}{53} a^{6} - \frac{9}{106} a^{5} + \frac{11}{53} a^{4} + \frac{9}{106} a^{3} + \frac{18}{53} a^{2} - \frac{89}{212} a - \frac{13}{106}$

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:  \( \frac{942}{53} a^{17} - \frac{8007}{53} a^{16} + \frac{32632}{53} a^{15} - \frac{84600}{53} a^{14} + \frac{169332}{53} a^{13} - \frac{303316}{53} a^{12} + \frac{489288}{53} a^{11} - \frac{650760}{53} a^{10} + \frac{710618}{53} a^{9} - \frac{707898}{53} a^{8} + \frac{669888}{53} a^{7} - \frac{534332}{53} a^{6} + \frac{340080}{53} a^{5} - \frac{190104}{53} a^{4} + \frac{99044}{53} a^{3} - \frac{40896}{53} a^{2} + \frac{10578}{53} a - \frac{1218}{53} \) (order $6$)
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:  \( 4456.90616557204 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.324.1, 3.1.108.1 x3, 6.0.314928.2, 6.0.34992.1, 9.1.14693280768.1 x3

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 6 sibling: data not computed
Degree 9 sibling: data not computed
Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.12.16.13$x^{12} + 12 x^{10} + 12 x^{8} + 8 x^{6} + 32 x^{4} - 16 x^{2} + 16$$6$$2$$16$$D_6$$[2]_{3}^{2}$
3Data not computed