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Properties

Label 18.6.48144624744...2096.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 3^{36}\cdot 23^{8}$
Root discriminant $57.56$
Ramified primes $2, 3, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3:S_4$ (as 18T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-288, 3456, -17280, 48672, -87048, 102312, -72978, 13770, 38484, -58939, 49905, -30510, 14172, -5292, 1584, -372, 72, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 72*x^16 - 372*x^15 + 1584*x^14 - 5292*x^13 + 14172*x^12 - 30510*x^11 + 49905*x^10 - 58939*x^9 + 38484*x^8 + 13770*x^7 - 72978*x^6 + 102312*x^5 - 87048*x^4 + 48672*x^3 - 17280*x^2 + 3456*x - 288)
 
gp: K = bnfinit(x^18 - 9*x^17 + 72*x^16 - 372*x^15 + 1584*x^14 - 5292*x^13 + 14172*x^12 - 30510*x^11 + 49905*x^10 - 58939*x^9 + 38484*x^8 + 13770*x^7 - 72978*x^6 + 102312*x^5 - 87048*x^4 + 48672*x^3 - 17280*x^2 + 3456*x - 288, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 72 x^{16} - 372 x^{15} + 1584 x^{14} - 5292 x^{13} + 14172 x^{12} - 30510 x^{11} + 49905 x^{10} - 58939 x^{9} + 38484 x^{8} + 13770 x^{7} - 72978 x^{6} + 102312 x^{5} - 87048 x^{4} + 48672 x^{3} - 17280 x^{2} + 3456 x - 288 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(48144624744449479590301442052096=2^{12}\cdot 3^{36}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.56$
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:  $2, 3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{12} a^{10} + \frac{1}{12} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{12} a^{6} - \frac{1}{12} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{11} - \frac{1}{4} a^{9} - \frac{1}{6} a^{8} - \frac{1}{4} a^{7} + \frac{1}{12} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{24} a^{12} - \frac{1}{24} a^{10} - \frac{1}{4} a^{9} + \frac{5}{24} a^{8} + \frac{1}{6} a^{7} - \frac{1}{24} a^{6} - \frac{1}{3} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{24} a^{13} - \frac{1}{24} a^{11} - \frac{1}{24} a^{9} + \frac{1}{6} a^{8} - \frac{1}{24} a^{7} - \frac{1}{12} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{48} a^{14} - \frac{1}{48} a^{13} - \frac{1}{48} a^{12} - \frac{1}{48} a^{11} - \frac{1}{48} a^{10} + \frac{11}{48} a^{9} - \frac{1}{48} a^{8} + \frac{5}{48} a^{7} - \frac{1}{12} a^{6} + \frac{11}{24} a^{5} - \frac{3}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{48} a^{15} + \frac{1}{8} a^{8} + \frac{1}{16} a^{7} - \frac{1}{12} a^{6} - \frac{1}{4} a^{5} - \frac{3}{8} a^{4} - \frac{1}{2} a$, $\frac{1}{46736736} a^{16} - \frac{1}{5842092} a^{15} - \frac{3605}{23368368} a^{14} + \frac{8435}{7789456} a^{13} - \frac{40547}{23368368} a^{12} - \frac{85865}{23368368} a^{11} + \frac{808135}{23368368} a^{10} - \frac{165625}{1460523} a^{9} + \frac{80485}{46736736} a^{8} + \frac{691861}{7789456} a^{7} - \frac{117971}{5842092} a^{6} - \frac{9105787}{23368368} a^{5} + \frac{1439861}{3894728} a^{4} - \frac{376377}{973682} a^{3} + \frac{391481}{1947364} a^{2} + \frac{212689}{973682} a - \frac{40492}{486841}$, $\frac{1}{1074944928} a^{17} + \frac{1}{358314976} a^{16} - \frac{213221}{33592029} a^{15} - \frac{816185}{179157488} a^{14} - \frac{9985853}{537472464} a^{13} + \frac{5797051}{537472464} a^{12} + \frac{9113599}{537472464} a^{11} - \frac{1475431}{89578744} a^{10} + \frac{20514457}{358314976} a^{9} - \frac{85515925}{1074944928} a^{8} - \frac{121745407}{537472464} a^{7} - \frac{107769983}{537472464} a^{6} - \frac{63287713}{537472464} a^{5} + \frac{40622377}{89578744} a^{4} - \frac{11003601}{22394686} a^{3} + \frac{15442171}{44789372} a^{2} - \frac{6504543}{22394686} a - \frac{2879617}{11197343}$

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:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 11618980288.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_4$ (as 18T37):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $C_3:S_4$
Character table for $C_3:S_4$

Intermediate fields

3.3.22356.3, 3.3.621.1, 3.3.22356.1, 3.3.22356.2, 6.2.499790736.1, 9.9.6938632771983936.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: 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 R R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{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$3.9.18.2$x^{9} + 3 x^{3} + 9 x^{2} + 9 x + 3$$9$$1$$18$$C_3^2:C_2$$[3/2, 5/2]_{2}$
3.9.18.2$x^{9} + 3 x^{3} + 9 x^{2} + 9 x + 3$$9$$1$$18$$C_3^2:C_2$$[3/2, 5/2]_{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$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}$