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Properties

Label 6.0.43243551.2
Degree $6$
Signature $[0, 3]$
Discriminant $-\,3^{9}\cdot 13^{3}$
Root discriminant $18.73$
Ramified primes $3, 13$
Class number $4$
Class group $[4]$
Galois group $S_3\times C_3$ (as 6T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12, -9, 9, -3, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 + 6*x^4 - 3*x^3 + 9*x^2 - 9*x + 12)
 
gp: K = bnfinit(x^6 + 6*x^4 - 3*x^3 + 9*x^2 - 9*x + 12, 1)
 

Normalized defining polynomial

\( x^{6} + 6 x^{4} - 3 x^{3} + 9 x^{2} - 9 x + 12 \)

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

Invariants

Degree:  $6$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-43243551=-\,3^{9}\cdot 13^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{10} a^{5} + \frac{3}{10} a^{4} - \frac{1}{2} a^{3} + \frac{1}{5} a^{2} - \frac{1}{2} a - \frac{2}{5}$

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

Class group and class number

$C_{4}$, which has order $4$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $2$
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:  \( \frac{6}{5} a^{5} + \frac{13}{5} a^{4} + 10 a^{3} + \frac{67}{5} a^{2} + 18 a + \frac{41}{5} \),  \( \frac{6}{5} a^{5} - \frac{2}{5} a^{4} + 2 a^{3} - \frac{13}{5} a^{2} + 3 a + \frac{1}{5} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 55.0494127817 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 6T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3\times C_3$
Character table for $S_3\times C_3$

Intermediate fields

\(\Q(\sqrt{-39}) \)

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

Sibling algebras

Galois closure: 18.0.6550117143611896491465231.1
Twin sextic algebra: \(\Q(\zeta_{9})^+\) $\times$ 3.1.351.1
Degree 9 sibling: 9.3.31524548679.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$3$3.6.9.15$x^{6} + 6 x^{4} + 6 x^{3} + 12$$6$$1$$9$$S_3\times C_3$$[3/2, 2]_{2}$
$13$13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3_13.2t1.1c1$1$ $ 3 \cdot 13 $ $x^{2} - x + 10$ $C_2$ (as 2T1) $1$ $-1$
1.3e2_13.6t1.3c1$1$ $ 3^{2} \cdot 13 $ $x^{6} + 18 x^{4} - 10 x^{3} + 81 x^{2} - 90 x + 181$ $C_6$ (as 6T1) $0$ $-1$
1.3e2_13.6t1.3c2$1$ $ 3^{2} \cdot 13 $ $x^{6} + 18 x^{4} - 10 x^{3} + 81 x^{2} - 90 x + 181$ $C_6$ (as 6T1) $0$ $-1$
1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
2.3e3_13.3t2.1c1$2$ $ 3^{3} \cdot 13 $ $x^{3} + 3 x - 3$ $S_3$ (as 3T2) $1$ $0$
* 2.3e4_13.6t5.1c1$2$ $ 3^{4} \cdot 13 $ $x^{6} + 6 x^{4} - 3 x^{3} + 9 x^{2} - 9 x + 12$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.3e4_13.6t5.1c2$2$ $ 3^{4} \cdot 13 $ $x^{6} + 6 x^{4} - 3 x^{3} + 9 x^{2} - 9 x + 12$ $S_3\times C_3$ (as 6T5) $0$ $0$

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 *.