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Properties

Label 18.0.11803922422...6272.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{37}$
Root discriminant $19.13$
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![4, 0, 18, -48, 117, -90, -30, -54, 189, -86, 27, -18, -15, 18, 9, -6, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^15 + 9*x^14 + 18*x^13 - 15*x^12 - 18*x^11 + 27*x^10 - 86*x^9 + 189*x^8 - 54*x^7 - 30*x^6 - 90*x^5 + 117*x^4 - 48*x^3 + 18*x^2 + 4)
 
gp: K = bnfinit(x^18 - 6*x^15 + 9*x^14 + 18*x^13 - 15*x^12 - 18*x^11 + 27*x^10 - 86*x^9 + 189*x^8 - 54*x^7 - 30*x^6 - 90*x^5 + 117*x^4 - 48*x^3 + 18*x^2 + 4, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{15} + 9 x^{14} + 18 x^{13} - 15 x^{12} - 18 x^{11} + 27 x^{10} - 86 x^{9} + 189 x^{8} - 54 x^{7} - 30 x^{6} - 90 x^{5} + 117 x^{4} - 48 x^{3} + 18 x^{2} + 4 \)

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:  \(-118039224225889612726272=-\,2^{18}\cdot 3^{37}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{18} a^{9} + \frac{1}{3} a^{6} - \frac{1}{6} a^{3} + \frac{1}{9}$, $\frac{1}{18} a^{10} - \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{9} a$, $\frac{1}{18} a^{11} - \frac{1}{6} a^{8} - \frac{1}{6} a^{5} - \frac{7}{18} a^{2}$, $\frac{1}{18} a^{12} - \frac{1}{6} a^{6} + \frac{1}{9} a^{3} + \frac{1}{3}$, $\frac{1}{54} a^{13} - \frac{1}{54} a^{12} - \frac{1}{54} a^{10} + \frac{1}{54} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{6} - \frac{1}{6} a^{5} + \frac{7}{27} a^{4} + \frac{2}{27} a^{3} + \frac{1}{3} a^{2} + \frac{2}{27} a + \frac{7}{27}$, $\frac{1}{108} a^{14} - \frac{1}{108} a^{12} + \frac{1}{54} a^{11} - \frac{1}{36} a^{10} - \frac{1}{54} a^{9} + \frac{1}{12} a^{8} + \frac{5}{12} a^{6} + \frac{25}{54} a^{5} + \frac{1}{4} a^{4} - \frac{25}{54} a^{3} - \frac{13}{54} a^{2} + \frac{1}{9} a - \frac{7}{27}$, $\frac{1}{108} a^{15} - \frac{1}{108} a^{13} + \frac{1}{54} a^{12} - \frac{1}{36} a^{11} - \frac{1}{54} a^{10} - \frac{1}{36} a^{9} - \frac{1}{12} a^{7} + \frac{8}{27} a^{6} - \frac{1}{4} a^{5} + \frac{1}{27} a^{4} - \frac{11}{27} a^{3} - \frac{7}{18} a^{2} - \frac{7}{27} a - \frac{2}{9}$, $\frac{1}{8856} a^{16} - \frac{1}{2952} a^{15} + \frac{7}{2952} a^{13} - \frac{109}{4428} a^{12} - \frac{1}{984} a^{11} + \frac{85}{4428} a^{10} + \frac{209}{8856} a^{9} + \frac{73}{492} a^{8} - \frac{661}{8856} a^{7} - \frac{133}{1476} a^{6} - \frac{391}{984} a^{5} - \frac{985}{2952} a^{4} - \frac{851}{4428} a^{3} + \frac{79}{164} a^{2} - \frac{899}{2214} a - \frac{911}{2214}$, $\frac{1}{97416} a^{17} - \frac{1}{32472} a^{16} + \frac{1}{297} a^{15} - \frac{13}{8856} a^{14} + \frac{73}{16236} a^{13} + \frac{1795}{97416} a^{12} + \frac{413}{48708} a^{11} + \frac{5}{10824} a^{10} - \frac{901}{48708} a^{9} + \frac{5243}{97416} a^{8} - \frac{379}{16236} a^{7} + \frac{39449}{97416} a^{6} - \frac{39199}{97416} a^{5} + \frac{263}{16236} a^{4} - \frac{11807}{48708} a^{3} - \frac{11149}{24354} a^{2} + \frac{1229}{2706} a + \frac{47}{297}$

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:  \( \frac{17}{2376} a^{17} - \frac{73}{2376} a^{16} + \frac{4}{297} a^{15} - \frac{5}{216} a^{14} + \frac{313}{1188} a^{13} - \frac{505}{2376} a^{12} - \frac{767}{1188} a^{11} + \frac{1601}{2376} a^{10} + \frac{1117}{1188} a^{9} - \frac{3929}{2376} a^{8} + \frac{4871}{1188} a^{7} - \frac{16823}{2376} a^{6} + \frac{5629}{2376} a^{5} + \frac{650}{297} a^{4} + \frac{4277}{1188} a^{3} - \frac{2687}{594} a^{2} + \frac{685}{594} a + \frac{89}{297} \) (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:  \( 160332.96342701954 \)
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

\(\Q(\sqrt{-3}) \), 3.3.1944.1, 3.1.243.1 x3, 6.0.11337408.1, 6.0.177147.2, 9.3.198359290368.1 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
3Data not computed