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Properties

Label 18.0.30616027031...1888.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 11^{14}$
Root discriminant $17.75$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![169, -234, 64, 134, -20, -216, 195, 2, -14, -36, 34, -38, 19, 8, -2, -2, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 2*x^16 - 2*x^15 - 2*x^14 + 8*x^13 + 19*x^12 - 38*x^11 + 34*x^10 - 36*x^9 - 14*x^8 + 2*x^7 + 195*x^6 - 216*x^5 - 20*x^4 + 134*x^3 + 64*x^2 - 234*x + 169)
 
gp: K = bnfinit(x^18 - 2*x^17 + 2*x^16 - 2*x^15 - 2*x^14 + 8*x^13 + 19*x^12 - 38*x^11 + 34*x^10 - 36*x^9 - 14*x^8 + 2*x^7 + 195*x^6 - 216*x^5 - 20*x^4 + 134*x^3 + 64*x^2 - 234*x + 169, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 2 x^{16} - 2 x^{15} - 2 x^{14} + 8 x^{13} + 19 x^{12} - 38 x^{11} + 34 x^{10} - 36 x^{9} - 14 x^{8} + 2 x^{7} + 195 x^{6} - 216 x^{5} - 20 x^{4} + 134 x^{3} + 64 x^{2} - 234 x + 169 \)

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:  \(-30616027031219947941888=-\,2^{12}\cdot 3^{9}\cdot 11^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.75$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{12} a^{12} + \frac{1}{12} a^{11} + \frac{1}{12} a^{10} - \frac{1}{12} a^{9} + \frac{1}{12} a^{8} - \frac{1}{12} a^{7} + \frac{1}{4} a^{6} - \frac{1}{12} a^{5} + \frac{5}{12} a^{4} - \frac{5}{12} a^{3} - \frac{5}{12} a^{2} - \frac{5}{12} a - \frac{1}{6}$, $\frac{1}{24} a^{13} - \frac{1}{8} a^{11} + \frac{1}{24} a^{10} + \frac{5}{24} a^{9} - \frac{5}{24} a^{8} - \frac{5}{24} a^{7} + \frac{5}{24} a^{6} + \frac{1}{8} a^{5} - \frac{1}{24} a^{4} - \frac{3}{8} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{24}$, $\frac{1}{24} a^{14} - \frac{1}{24} a^{12} - \frac{1}{8} a^{11} + \frac{1}{24} a^{10} - \frac{1}{24} a^{9} + \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{3}{8} a^{6} + \frac{1}{8} a^{5} - \frac{5}{24} a^{4} + \frac{5}{24} a^{3} - \frac{1}{6} a^{2} + \frac{7}{24} a - \frac{5}{12}$, $\frac{1}{24} a^{15} - \frac{1}{24} a^{12} + \frac{1}{12} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{6} a^{7} + \frac{1}{12} a^{6} + \frac{1}{3} a^{5} + \frac{1}{12} a^{4} + \frac{1}{24} a^{3} + \frac{1}{4} a^{2} - \frac{1}{3} a - \frac{5}{24}$, $\frac{1}{2064} a^{16} - \frac{1}{516} a^{15} - \frac{37}{2064} a^{14} + \frac{1}{1032} a^{13} - \frac{73}{2064} a^{12} + \frac{73}{1032} a^{11} + \frac{39}{344} a^{10} - \frac{61}{258} a^{9} - \frac{23}{258} a^{8} - \frac{39}{172} a^{7} + \frac{133}{344} a^{6} - \frac{11}{344} a^{5} + \frac{661}{2064} a^{4} + \frac{73}{172} a^{3} + \frac{701}{2064} a^{2} + \frac{409}{1032} a - \frac{183}{688}$, $\frac{1}{192586626336} a^{17} - \frac{20061355}{192586626336} a^{16} + \frac{1189203891}{64195542112} a^{15} - \frac{962535553}{64195542112} a^{14} - \frac{1167680815}{192586626336} a^{13} - \frac{2157525869}{192586626336} a^{12} - \frac{2424667033}{24073328292} a^{11} + \frac{162127453}{96293313168} a^{10} - \frac{318783921}{2006110691} a^{9} + \frac{11914272361}{48146656584} a^{8} - \frac{9783805129}{96293313168} a^{7} - \frac{22177615765}{48146656584} a^{6} - \frac{6921257149}{14814355872} a^{5} + \frac{92551584689}{192586626336} a^{4} + \frac{26793381149}{64195542112} a^{3} + \frac{11155549487}{192586626336} a^{2} + \frac{18491830699}{64195542112} a - \frac{677799611}{14814355872}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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:  \( -\frac{2133979}{746459792} a^{17} - \frac{135584}{139961211} a^{16} + \frac{4126585}{2239379376} a^{15} - \frac{330795}{373229896} a^{14} + \frac{13121309}{746459792} a^{13} - \frac{8056019}{1119689688} a^{12} - \frac{49909555}{559844844} a^{11} - \frac{24338881}{373229896} a^{10} + \frac{14311807}{1119689688} a^{9} - \frac{500485}{373229896} a^{8} + \frac{152335169}{559844844} a^{7} + \frac{34539393}{186614948} a^{6} - \frac{75029111}{172259952} a^{5} - \frac{838031873}{1119689688} a^{4} + \frac{814419409}{2239379376} a^{3} - \frac{30164275}{279922422} a^{2} - \frac{41689795}{2239379376} a + \frac{12480559}{28709992} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 39498.29051696313 \) (assuming GRH)
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

\(\Q(\sqrt{-3}) \), 3.1.44.1, 3.1.1452.1 x3, 6.0.52272.1, 6.0.6324912.1, 9.1.33673831488.1 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$11$11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.12.10.1$x^{12} + 3146 x^{6} + 14235529$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$