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Properties

Label 18.0.63426874804...4832.6
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 23^{6}$
Root discriminant $16.26$
Ramified primes $2, 3, 23$
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, -48, 252, -760, 1482, -2058, 2267, -2109, 1557, -790, 225, -21, 44, -69, 51, -22, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 9*x^16 - 22*x^15 + 51*x^14 - 69*x^13 + 44*x^12 - 21*x^11 + 225*x^10 - 790*x^9 + 1557*x^8 - 2109*x^7 + 2267*x^6 - 2058*x^5 + 1482*x^4 - 760*x^3 + 252*x^2 - 48*x + 4)
 
gp: K = bnfinit(x^18 - 3*x^17 + 9*x^16 - 22*x^15 + 51*x^14 - 69*x^13 + 44*x^12 - 21*x^11 + 225*x^10 - 790*x^9 + 1557*x^8 - 2109*x^7 + 2267*x^6 - 2058*x^5 + 1482*x^4 - 760*x^3 + 252*x^2 - 48*x + 4, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 9 x^{16} - 22 x^{15} + 51 x^{14} - 69 x^{13} + 44 x^{12} - 21 x^{11} + 225 x^{10} - 790 x^{9} + 1557 x^{8} - 2109 x^{7} + 2267 x^{6} - 2058 x^{5} + 1482 x^{4} - 760 x^{3} + 252 x^{2} - 48 x + 4 \)

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:  \(-6342687480463779704832=-\,2^{12}\cdot 3^{21}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.26$
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:  $2, 3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{9} + \frac{1}{6} a^{6} - \frac{1}{2} a^{3} - \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{2} a^{10} + \frac{1}{6} a^{7} - \frac{1}{2} a^{4} - \frac{1}{3} a$, $\frac{1}{6} a^{14} - \frac{1}{2} a^{11} + \frac{1}{6} a^{8} - \frac{1}{2} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{18} a^{15} + \frac{1}{18} a^{12} - \frac{1}{3} a^{10} - \frac{5}{18} a^{9} + \frac{1}{3} a^{7} - \frac{5}{18} a^{6} - \frac{1}{3} a^{5} - \frac{4}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{4}{9}$, $\frac{1}{108} a^{16} - \frac{1}{54} a^{15} - \frac{1}{36} a^{14} + \frac{7}{108} a^{13} - \frac{1}{54} a^{12} + \frac{7}{36} a^{11} + \frac{7}{108} a^{10} + \frac{23}{54} a^{9} - \frac{11}{36} a^{8} + \frac{7}{108} a^{7} - \frac{8}{27} a^{6} - \frac{11}{36} a^{5} + \frac{7}{27} a^{4} + \frac{5}{54} a^{3} - \frac{5}{18} a^{2} + \frac{10}{27} a + \frac{13}{27}$, $\frac{1}{33531226452} a^{17} - \frac{757513}{620948638} a^{16} + \frac{450039365}{33531226452} a^{15} - \frac{1190556305}{33531226452} a^{14} - \frac{101393537}{2794268871} a^{13} - \frac{2322781591}{33531226452} a^{12} + \frac{4188661627}{33531226452} a^{11} - \frac{10686797}{96354099} a^{10} - \frac{4943475205}{33531226452} a^{9} + \frac{2750454139}{33531226452} a^{8} - \frac{288736651}{620948638} a^{7} + \frac{1931548643}{33531226452} a^{6} + \frac{2398623241}{8382806613} a^{5} + \frac{1324943632}{2794268871} a^{4} + \frac{6881779015}{16765613226} a^{3} + \frac{2545685806}{8382806613} a^{2} + \frac{1371790610}{2794268871} a - \frac{378681997}{8382806613}$

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:  \( -\frac{6169751}{635013} a^{17} + \frac{59639741}{2540052} a^{16} - \frac{46559291}{635013} a^{15} + \frac{431877679}{2540052} a^{14} - \frac{999007069}{2540052} a^{13} + \frac{275650885}{635013} a^{12} - \frac{402326753}{2540052} a^{11} + \frac{8384575}{87588} a^{10} - \frac{1350469760}{635013} a^{9} + \frac{16336647265}{2540052} a^{8} - \frac{28659425533}{2540052} a^{7} + \frac{17328551765}{1270026} a^{6} - \frac{34605897167}{2540052} a^{5} + \frac{7326946733}{635013} a^{4} - \frac{9112564255}{1270026} a^{3} + \frac{3548886995}{1270026} a^{2} - \frac{373917613}{635013} a + \frac{32223454}{635013} \) (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:  \( 34377.40711821429 \)
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

\(\Q(\sqrt{-3}) \), 3.3.621.1, 3.1.108.1 x3, 6.0.1156923.1, 6.0.34992.1, 9.3.45980747712.1 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.4.1$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$$[\ ]_{3}^{2}$
$3$3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$