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Properties

Label 18.6.60180780930...2000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 3^{36}\cdot 5^{3}\cdot 23^{8}$
Root discriminant $75.27$
Ramified primes $2, 3, 5, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_3:S_4$ (as 18T66)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8000, 0, -57600, 0, -34560, 0, 168816, 0, 112320, 0, 17280, 0, -1020, 0, -288, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 288*x^14 - 1020*x^12 + 17280*x^10 + 112320*x^8 + 168816*x^6 - 34560*x^4 - 57600*x^2 - 8000)
 
gp: K = bnfinit(x^18 - 288*x^14 - 1020*x^12 + 17280*x^10 + 112320*x^8 + 168816*x^6 - 34560*x^4 - 57600*x^2 - 8000, 1)
 

Normalized defining polynomial

\( x^{18} - 288 x^{14} - 1020 x^{12} + 17280 x^{10} + 112320 x^{8} + 168816 x^{6} - 34560 x^{4} - 57600 x^{2} - 8000 \)

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:  \(6018078093056184948787680256512000=2^{12}\cdot 3^{36}\cdot 5^{3}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.27$
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:  $2, 3, 5, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6}$, $\frac{1}{8} a^{7} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{6} - \frac{1}{4} a^{3} - \frac{1}{2}$, $\frac{1}{32} a^{10} - \frac{1}{16} a^{7} - \frac{1}{8} a^{4} - \frac{1}{4} a$, $\frac{1}{32} a^{11} - \frac{1}{16} a^{8} - \frac{1}{8} a^{5} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{12} - \frac{1}{8} a^{6} - \frac{1}{4} a^{3} - \frac{1}{4}$, $\frac{1}{64} a^{13} - \frac{1}{4} a^{4} + \frac{1}{4} a$, $\frac{1}{7360} a^{14} + \frac{7}{1472} a^{12} - \frac{11}{920} a^{10} + \frac{1}{46} a^{8} + \frac{21}{184} a^{6} - \frac{7}{46} a^{4} - \frac{1}{4} a^{3} - \frac{229}{460} a^{2} + \frac{33}{92}$, $\frac{1}{14720} a^{15} - \frac{1}{184} a^{13} - \frac{11}{1840} a^{11} + \frac{1}{92} a^{9} - \frac{1}{184} a^{7} - \frac{7}{92} a^{5} + \frac{1}{920} a^{3} - \frac{9}{46} a - \frac{1}{2}$, $\frac{1}{65788271830400} a^{16} + \frac{290716327}{6578827183040} a^{14} - \frac{87455596469}{32894135915200} a^{12} + \frac{6984721853}{1644706795760} a^{10} - \frac{317089479}{5107785080} a^{8} + \frac{52228170089}{822353397880} a^{6} - \frac{822165981149}{4111766989400} a^{4} - \frac{1}{4} a^{3} - \frac{108432974269}{411176698940} a^{2} - \frac{1}{2} a + \frac{7883975133}{82235339788}$, $\frac{1}{131576543660800} a^{17} + \frac{290716327}{13157654366080} a^{15} - \frac{1}{14720} a^{14} + \frac{213257638603}{32894135915200} a^{13} + \frac{1}{184} a^{12} + \frac{6984721853}{3289413591520} a^{11} + \frac{11}{1840} a^{10} - \frac{317089479}{10215570160} a^{9} - \frac{1}{92} a^{8} - \frac{25283002323}{822353397880} a^{7} + \frac{1}{184} a^{6} + \frac{205775766201}{8223533978800} a^{5} - \frac{4}{23} a^{4} + \frac{97155375201}{822353397880} a^{3} + \frac{229}{920} a^{2} - \frac{6337429907}{82235339788} a - \frac{7}{23}$

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:  \( 23860123501.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_4$ (as 18T66):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 18 conjugacy class representatives for $C_2\times C_3:S_4$
Character table for $C_2\times C_3:S_4$

Intermediate fields

3.3.22356.1, 3.3.22356.3, 3.3.22356.2, 3.3.621.1, 6.2.2498953680.2, 9.9.6938632771983936.1

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

Sibling fields

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 R ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.6.4.1$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$$[\ ]_{3}^{2}$
3Data not computed
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$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}$