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Properties

Label 18.2.53057740732...4624.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{33}\cdot 3^{31}$
Root discriminant $23.64$
Ramified primes $2, 3$
Class number $3$
Class group $[3]$
Galois group $S_3^2$ (as 18T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-125, 0, 0, -798, 0, 0, 555, 0, 0, -164, 0, 0, 51, 0, 0, -6, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^15 + 51*x^12 - 164*x^9 + 555*x^6 - 798*x^3 - 125)
 
gp: K = bnfinit(x^18 - 6*x^15 + 51*x^12 - 164*x^9 + 555*x^6 - 798*x^3 - 125, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{15} + 51 x^{12} - 164 x^{9} + 555 x^{6} - 798 x^{3} - 125 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5305774073297600589594624=2^{33}\cdot 3^{31}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.64$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{1}{9} a^{4} + \frac{1}{9} a^{3} - \frac{2}{9} a^{2} - \frac{2}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{3} + \frac{2}{9}$, $\frac{1}{45} a^{10} + \frac{2}{15} a^{7} + \frac{2}{15} a^{4} + \frac{1}{3} a^{3} - \frac{4}{45} a - \frac{1}{3}$, $\frac{1}{135} a^{11} + \frac{1}{135} a^{10} - \frac{1}{27} a^{9} + \frac{2}{45} a^{8} + \frac{2}{45} a^{7} + \frac{1}{9} a^{6} + \frac{7}{45} a^{5} + \frac{7}{45} a^{4} - \frac{4}{9} a^{3} + \frac{26}{135} a^{2} + \frac{26}{135} a + \frac{10}{27}$, $\frac{1}{135} a^{12} - \frac{4}{135} a^{9} + \frac{2}{45} a^{6} - \frac{4}{135} a^{3} + \frac{11}{27}$, $\frac{1}{135} a^{13} - \frac{1}{135} a^{10} - \frac{7}{45} a^{7} + \frac{14}{135} a^{4} - \frac{47}{135} a$, $\frac{1}{135} a^{14} + \frac{1}{135} a^{10} - \frac{1}{27} a^{9} + \frac{7}{45} a^{7} - \frac{1}{9} a^{6} + \frac{1}{27} a^{5} - \frac{1}{15} a^{4} - \frac{2}{45} a^{2} + \frac{41}{135} a + \frac{4}{27}$, $\frac{1}{405} a^{15} + \frac{1}{405} a^{12} - \frac{14}{405} a^{9} + \frac{26}{405} a^{6} - \frac{20}{81} a^{3} + \frac{1}{81}$, $\frac{1}{405} a^{16} + \frac{1}{405} a^{13} + \frac{4}{405} a^{10} - \frac{1}{405} a^{7} + \frac{8}{405} a^{4} + \frac{1}{3} a^{3} + \frac{68}{405} a - \frac{1}{3}$, $\frac{1}{2025} a^{17} + \frac{4}{2025} a^{14} + \frac{1}{2025} a^{11} - \frac{64}{2025} a^{8} - \frac{44}{405} a^{5} - \frac{208}{2025} a^{2}$

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

Class group and class number

$C_{3}$, which has order $3$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 198871.4244922057 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{6}) \), 3.1.108.1, 3.1.648.1, 6.2.40310784.5 x2, 6.2.4478976.4, 6.2.10077696.2, 9.1.117546246144.4

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

Sibling fields

Galois closure: data not computed
Degree 6 sibling: data not computed
Degree 9 sibling: data not computed
Degree 12 sibling: 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.11.13$x^{6} + 10$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.12.22.79$x^{12} + 2 x^{10} + 4 x^{8} + 4 x^{6} + 4 x^{4} + 4$$6$$2$$22$$D_6$$[3]_{3}^{2}$
3Data not computed