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Properties

Label 18.0.38604587267...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{8}\cdot 3^{31}\cdot 5^{12}$
Root discriminant $26.39$
Ramified primes $2, 3, 5$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
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![4096, 0, 0, -5376, 0, 0, 3120, 0, 0, -340, 0, 0, 195, 0, 0, -21, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 21*x^15 + 195*x^12 - 340*x^9 + 3120*x^6 - 5376*x^3 + 4096)
 
gp: K = bnfinit(x^18 - 21*x^15 + 195*x^12 - 340*x^9 + 3120*x^6 - 5376*x^3 + 4096, 1)
 

Normalized defining polynomial

\( x^{18} - 21 x^{15} + 195 x^{12} - 340 x^{9} + 3120 x^{6} - 5376 x^{3} + 4096 \)

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:  \(-38604587267746687500000000=-\,2^{8}\cdot 3^{31}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.39$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{5} + \frac{1}{6} a^{2}$, $\frac{1}{60} a^{9} + \frac{7}{60} a^{6} + \frac{23}{60} a^{3} - \frac{4}{15}$, $\frac{1}{180} a^{10} + \frac{1}{180} a^{9} + \frac{3}{20} a^{7} + \frac{3}{20} a^{6} + \frac{7}{20} a^{4} + \frac{7}{20} a^{3} + \frac{16}{45} a + \frac{16}{45}$, $\frac{1}{360} a^{11} + \frac{1}{180} a^{9} + \frac{3}{40} a^{8} + \frac{3}{20} a^{6} - \frac{13}{40} a^{5} + \frac{7}{20} a^{3} - \frac{29}{90} a^{2} + \frac{16}{45}$, $\frac{1}{720} a^{12} - \frac{1}{144} a^{9} + \frac{11}{80} a^{6} + \frac{7}{180} a^{3} + \frac{7}{45}$, $\frac{1}{4320} a^{13} - \frac{1}{2160} a^{12} - \frac{1}{864} a^{10} - \frac{7}{2160} a^{9} - \frac{47}{1440} a^{7} - \frac{61}{720} a^{6} + \frac{1}{3} a^{5} - \frac{473}{1080} a^{4} + \frac{26}{135} a^{3} + \frac{1}{3} a^{2} + \frac{127}{270} a + \frac{10}{27}$, $\frac{1}{8640} a^{14} - \frac{1}{2160} a^{12} - \frac{1}{1728} a^{11} + \frac{17}{2160} a^{9} - \frac{47}{2880} a^{8} - \frac{17}{144} a^{6} + \frac{247}{2160} a^{5} + \frac{1}{3} a^{4} - \frac{119}{270} a^{3} - \frac{233}{540} a^{2} + \frac{1}{3} a + \frac{11}{135}$, $\frac{1}{449280} a^{15} - \frac{277}{449280} a^{12} + \frac{1219}{449280} a^{9} - \frac{2641}{22464} a^{6} - \frac{10189}{28080} a^{3} - \frac{719}{1755}$, $\frac{1}{898560} a^{16} - \frac{23}{299520} a^{13} - \frac{1}{2160} a^{12} + \frac{179}{898560} a^{10} - \frac{7}{2160} a^{9} + \frac{16903}{224640} a^{7} - \frac{61}{720} a^{6} + \frac{1}{3} a^{5} - \frac{119}{416} a^{4} + \frac{26}{135} a^{3} + \frac{1}{3} a^{2} - \frac{119}{1755} a + \frac{10}{27}$, $\frac{1}{1797120} a^{17} - \frac{23}{599040} a^{14} - \frac{1}{2160} a^{12} + \frac{179}{1797120} a^{11} + \frac{17}{2160} a^{9} + \frac{16903}{449280} a^{8} - \frac{17}{144} a^{6} - \frac{773}{2496} a^{5} + \frac{1}{3} a^{4} - \frac{119}{270} a^{3} - \frac{352}{1755} a^{2} + \frac{1}{3} a + \frac{11}{135}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( \frac{17}{49920} a^{15} - \frac{341}{49920} a^{12} + \frac{3043}{49920} a^{9} - \frac{937}{12480} a^{6} + \frac{3691}{3120} a^{3} - \frac{107}{195} \) (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:  \( 528671.2860542445 \) (assuming GRH)
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

\(\Q(\sqrt{-3}) \), 3.1.675.1 x3, 6.0.1366875.1, 9.3.3587226750000.3

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 ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
3Data not computed
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$