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Properties

Label 6.0.419904.3
Degree $6$
Signature $[0, 3]$
Discriminant $-\,2^{6}\cdot 3^{8}$
Root discriminant $8.65$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
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![5, -12, 9, -2, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^3 + 9*x^2 - 12*x + 5)
 
gp: K = bnfinit(x^6 - 2*x^3 + 9*x^2 - 12*x + 5, 1)
 

Normalized defining polynomial

\( x^{6} - 2 x^{3} + 9 x^{2} - 12 x + 5 \)

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:  \(-419904=-\,2^{6}\cdot 3^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $8.65$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{19} a^{5} + \frac{7}{19} a^{4} - \frac{8}{19} a^{3} - \frac{1}{19} a^{2} + \frac{2}{19} a + \frac{2}{19}$

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:  $2$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{9}{19} a^{5} - \frac{6}{19} a^{4} - \frac{4}{19} a^{3} + \frac{9}{19} a^{2} - \frac{75}{19} a + \frac{58}{19} \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( \frac{9}{19} a^{5} + \frac{6}{19} a^{4} + \frac{4}{19} a^{3} - \frac{9}{19} a^{2} + \frac{56}{19} a - \frac{39}{19} \),  \( \frac{3}{19} a^{5} + \frac{2}{19} a^{4} - \frac{5}{19} a^{3} - \frac{3}{19} a^{2} + \frac{25}{19} a - \frac{13}{19} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 5.28645643215 \)
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

\(\Q(\sqrt{-1}) \)

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

Sibling algebras

Galois closure: 18.0.485758124386377007104.1
Twin sextic algebra: \(\Q(\zeta_{9})^+\) $\times$ 3.1.324.1
Degree 9 sibling: 9.3.2754990144.1

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 ${\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }$

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$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.6.8.9$x^{6} + 6 x^{5} + 9$$3$$2$$8$$S_3\times C_3$$[2, 2]^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e2.2t1.1c1$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.2e2_3e2.6t1.2c1$1$ $ 2^{2} \cdot 3^{2}$ $x^{6} + 6 x^{4} + 9 x^{2} + 1$ $C_6$ (as 6T1) $0$ $-1$
1.2e2_3e2.6t1.2c2$1$ $ 2^{2} \cdot 3^{2}$ $x^{6} + 6 x^{4} + 9 x^{2} + 1$ $C_6$ (as 6T1) $0$ $-1$
1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
2.2e2_3e4.3t2.1c1$2$ $ 2^{2} \cdot 3^{4}$ $x^{3} - 3 x - 4$ $S_3$ (as 3T2) $1$ $0$
* 2.2e2_3e4.6t5.2c1$2$ $ 2^{2} \cdot 3^{4}$ $x^{6} - 2 x^{3} + 9 x^{2} - 12 x + 5$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2e2_3e4.6t5.2c2$2$ $ 2^{2} \cdot 3^{4}$ $x^{6} - 2 x^{3} + 9 x^{2} - 12 x + 5$ $S_3\times C_3$ (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 *.