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Properties

Label 6.6.187388721.1
Degree $6$
Signature $[6, 0]$
Discriminant $3^{8}\cdot 13^{4}$
Root discriminant $23.92$
Ramified primes $3, 13$
Class number $1$
Class group Trivial
Galois group $C_3^2:C_4$ (as 6T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9, 9, 27, -9, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 15*x^4 - 9*x^3 + 27*x^2 + 9*x - 9)
 
gp: K = bnfinit(x^6 - 15*x^4 - 9*x^3 + 27*x^2 + 9*x - 9, 1)
 

Normalized defining polynomial

\( x^{6} - 15 x^{4} - 9 x^{3} + 27 x^{2} + 9 x - 9 \)

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

Invariants

Degree:  $6$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(187388721=3^{8}\cdot 13^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.92$
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:  $3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{3} a^{4}$, $\frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$

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:  $5$
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:  \( \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{7}{3} a^{3} + \frac{1}{2} a^{2} + 4 a + \frac{3}{2} \),  \( \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{8}{3} a^{3} - \frac{7}{2} a^{2} + 4 a + \frac{9}{2} \),  \( \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{7}{3} a^{3} - \frac{7}{2} a^{2} + a + \frac{3}{2} \),  \( \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - 2 a^{3} + \frac{1}{2} a^{2} + a - \frac{1}{2} \),  \( \frac{1}{3} a^{5} - a^{4} - 5 a^{3} + 11 a^{2} + 18 a - 11 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 278.629803956 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3.C_2$ (as 6T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 6 conjugacy class representatives for $C_3^2:C_4$
Character table for $C_3^2:C_4$

Intermediate fields

\(\Q(\sqrt{13}) \)

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

Sibling algebras

Galois closure: data not computed
Twin sextic algebra: 6.6.1686498489.1
Degree 6 sibling: 6.6.1686498489.1
Degree 9 sibling: 9.9.1870004703089601.1
Degree 12 siblings: Deg 12, Deg 12
Degree 18 sibling: 18.18.45459928664503948293335446409613.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$3$3.3.5.2$x^{3} + 21$$3$$1$$5$$S_3$$[5/2]_{2}$
3.3.3.2$x^{3} + 3 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.13.2t1.1c1$1$ $ 13 $ $x^{2} - x - 3$ $C_2$ (as 2T1) $1$ $1$
1.3_13.4t1.1c1$1$ $ 3 \cdot 13 $ $x^{4} - x^{3} - 11 x^{2} - 9 x + 3$ $C_4$ (as 4T1) $0$ $1$
1.3_13.4t1.1c2$1$ $ 3 \cdot 13 $ $x^{4} - x^{3} - 11 x^{2} - 9 x + 3$ $C_4$ (as 4T1) $0$ $1$
* 4.3e8_13e3.6t10.1c1$4$ $ 3^{8} \cdot 13^{3}$ $x^{6} - 15 x^{4} - 9 x^{3} + 27 x^{2} + 9 x - 9$ $C_3^2:C_4$ (as 6T10) $1$ $4$
4.3e10_13e3.6t10.1c1$4$ $ 3^{10} \cdot 13^{3}$ $x^{6} - 15 x^{4} - 9 x^{3} + 27 x^{2} + 9 x - 9$ $C_3^2:C_4$ (as 6T10) $1$ $4$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.