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Properties

Label 18.0.16736565124...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 5^{8}$
Root discriminant $11.69$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 3, -1, 9, -15, 37, -45, 69, -92, 126, -129, 109, -69, 39, -18, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 9*x^16 - 18*x^15 + 39*x^14 - 69*x^13 + 109*x^12 - 129*x^11 + 126*x^10 - 92*x^9 + 69*x^8 - 45*x^7 + 37*x^6 - 15*x^5 + 9*x^4 - x^3 + 3*x^2 + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 + 9*x^16 - 18*x^15 + 39*x^14 - 69*x^13 + 109*x^12 - 129*x^11 + 126*x^10 - 92*x^9 + 69*x^8 - 45*x^7 + 37*x^6 - 15*x^5 + 9*x^4 - x^3 + 3*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 9 x^{16} - 18 x^{15} + 39 x^{14} - 69 x^{13} + 109 x^{12} - 129 x^{11} + 126 x^{10} - 92 x^{9} + 69 x^{8} - 45 x^{7} + 37 x^{6} - 15 x^{5} + 9 x^{4} - x^{3} + 3 x^{2} + 1 \)

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:  \(-16736565124800000000=-\,2^{12}\cdot 3^{21}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.69$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{35} a^{15} - \frac{1}{35} a^{14} + \frac{9}{35} a^{13} - \frac{8}{35} a^{12} + \frac{12}{35} a^{11} - \frac{2}{7} a^{10} - \frac{2}{5} a^{9} - \frac{4}{35} a^{8} + \frac{2}{7} a^{7} + \frac{4}{35} a^{6} + \frac{16}{35} a^{5} + \frac{1}{7} a^{4} - \frac{3}{7} a^{3} + \frac{9}{35} a^{2} + \frac{2}{35} a + \frac{6}{35}$, $\frac{1}{35} a^{16} + \frac{8}{35} a^{14} + \frac{1}{35} a^{13} + \frac{4}{35} a^{12} + \frac{2}{35} a^{11} + \frac{11}{35} a^{10} + \frac{17}{35} a^{9} + \frac{6}{35} a^{8} + \frac{2}{5} a^{7} - \frac{3}{7} a^{6} - \frac{2}{5} a^{5} - \frac{2}{7} a^{4} - \frac{6}{35} a^{3} + \frac{11}{35} a^{2} + \frac{8}{35} a + \frac{6}{35}$, $\frac{1}{24955} a^{17} - \frac{2}{1085} a^{16} - \frac{152}{24955} a^{15} + \frac{10083}{24955} a^{14} - \frac{5017}{24955} a^{13} - \frac{3942}{24955} a^{12} + \frac{7764}{24955} a^{11} + \frac{3981}{24955} a^{10} - \frac{5641}{24955} a^{9} + \frac{924}{3565} a^{8} - \frac{5689}{24955} a^{7} + \frac{32}{1085} a^{6} - \frac{10221}{24955} a^{5} + \frac{2419}{24955} a^{4} + \frac{10072}{24955} a^{3} + \frac{407}{24955} a^{2} - \frac{8942}{24955} a + \frac{199}{24955}$

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:  \( \frac{42}{713} a^{17} - \frac{17}{155} a^{16} + \frac{878}{3565} a^{15} - \frac{1606}{3565} a^{14} + \frac{3096}{3565} a^{13} - \frac{5018}{3565} a^{12} + \frac{3374}{3565} a^{11} + \frac{374}{3565} a^{10} - \frac{6734}{3565} a^{9} + \frac{9997}{3565} a^{8} - \frac{16814}{3565} a^{7} + \frac{954}{155} a^{6} - \frac{25948}{3565} a^{5} + \frac{2491}{713} a^{4} - \frac{8194}{3565} a^{3} + \frac{1336}{3565} a^{2} - \frac{5482}{3565} a + \frac{1862}{3565} \) (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:  \( 302.516517627 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, 6.0.34992.1, 9.3.787320000.1

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

Sibling fields

Degree 9 siblings: 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 R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
3Data not computed
$5$5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$