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Properties

Label 18.0.88944969064...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{33}\cdot 5^{8}$
Root discriminant $24.33$
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![28, 108, 156, 228, 366, 114, -107, 150, -243, -328, 462, 114, -275, 72, 30, -26, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 15*x^16 - 26*x^15 + 30*x^14 + 72*x^13 - 275*x^12 + 114*x^11 + 462*x^10 - 328*x^9 - 243*x^8 + 150*x^7 - 107*x^6 + 114*x^5 + 366*x^4 + 228*x^3 + 156*x^2 + 108*x + 28)
 
gp: K = bnfinit(x^18 - 6*x^17 + 15*x^16 - 26*x^15 + 30*x^14 + 72*x^13 - 275*x^12 + 114*x^11 + 462*x^10 - 328*x^9 - 243*x^8 + 150*x^7 - 107*x^6 + 114*x^5 + 366*x^4 + 228*x^3 + 156*x^2 + 108*x + 28, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 15 x^{16} - 26 x^{15} + 30 x^{14} + 72 x^{13} - 275 x^{12} + 114 x^{11} + 462 x^{10} - 328 x^{9} - 243 x^{8} + 150 x^{7} - 107 x^{6} + 114 x^{5} + 366 x^{4} + 228 x^{3} + 156 x^{2} + 108 x + 28 \)

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:  \(-8894496906488836800000000=-\,2^{12}\cdot 3^{33}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.33$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{21892} a^{16} + \frac{73}{21892} a^{15} - \frac{81}{1684} a^{14} - \frac{1015}{10946} a^{13} - \frac{1363}{21892} a^{12} - \frac{2623}{21892} a^{11} + \frac{1541}{21892} a^{10} + \frac{2}{421} a^{9} - \frac{4951}{21892} a^{8} + \frac{3249}{21892} a^{7} - \frac{277}{21892} a^{6} + \frac{103}{842} a^{5} + \frac{1559}{10946} a^{4} + \frac{1603}{10946} a^{3} - \frac{2183}{5473} a^{2} + \frac{220}{5473} a + \frac{2407}{5473}$, $\frac{1}{21783243368068} a^{17} + \frac{70756011}{21783243368068} a^{16} + \frac{366895104679}{5445810842017} a^{15} + \frac{5017724562017}{21783243368068} a^{14} - \frac{278676132198}{5445810842017} a^{13} - \frac{498841094893}{21783243368068} a^{12} + \frac{1927609533557}{10891621684034} a^{11} + \frac{1678573351939}{21783243368068} a^{10} - \frac{704472534473}{5445810842017} a^{9} + \frac{10104699313915}{21783243368068} a^{8} + \frac{1170996251104}{5445810842017} a^{7} - \frac{2347092715953}{21783243368068} a^{6} - \frac{1966021951571}{21783243368068} a^{5} + \frac{2077134048664}{5445810842017} a^{4} + \frac{117890797879}{418908526309} a^{3} - \frac{3341848946881}{10891621684034} a^{2} - \frac{551713726048}{5445810842017} a - \frac{160510647403}{777972977431}$

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:  \( \frac{679228677}{9206780798} a^{17} - \frac{4871431341}{9206780798} a^{16} + \frac{31586491031}{18413561596} a^{15} - \frac{70673298813}{18413561596} a^{14} + \frac{117245319255}{18413561596} a^{13} - \frac{11509977063}{9206780798} a^{12} - \frac{378459892833}{18413561596} a^{11} + \frac{626289219891}{18413561596} a^{10} - \frac{64800736109}{18413561596} a^{9} - \frac{262111115385}{9206780798} a^{8} + \frac{451095265419}{18413561596} a^{7} - \frac{318508527811}{18413561596} a^{6} - \frac{8084700495}{18413561596} a^{5} + \frac{197806258401}{9206780798} a^{4} + \frac{10672559588}{4603390399} a^{3} + \frac{65580192081}{9206780798} a^{2} + \frac{23651545230}{4603390399} a + \frac{653365325}{4603390399} \) (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:  \( 960741.0373621208 \)
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

\(\Q(\sqrt{-3}) \), 3.1.972.2 x3, 6.0.2834352.2, 9.3.573956280000.2

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 ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.4.1$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$$[\ ]_{3}^{2}$
3Data not computed
$5$5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
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}$