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Properties

Label 6.0.97367427.2
Degree $6$
Signature $[0, 3]$
Discriminant $-\,3^{7}\cdot 211^{2}$
Root discriminant $21.45$
Ramified primes $3, 211$
Class number $3$
Class group $[3]$
Galois group $S_3^2$ (as 6T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![57, 162, 141, 27, -12, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 - 12*x^4 + 27*x^3 + 141*x^2 + 162*x + 57)
 
gp: K = bnfinit(x^6 - 3*x^5 - 12*x^4 + 27*x^3 + 141*x^2 + 162*x + 57, 1)
 

Normalized defining polynomial

\( x^{6} - 3 x^{5} - 12 x^{4} + 27 x^{3} + 141 x^{2} + 162 x + 57 \)

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:  \(-97367427=-\,3^{7}\cdot 211^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.45$
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:  $3, 211$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{119} a^{5} - \frac{47}{119} a^{4} + \frac{33}{119} a^{3} + \frac{3}{119} a^{2} + \frac{9}{119} a + \frac{4}{119}$

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:  $2$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{80}{119} a^{5} - \frac{309}{119} a^{4} - \frac{692}{119} a^{3} + \frac{2739}{119} a^{2} + \frac{8931}{119} a + \frac{5437}{119} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( \frac{11}{119} a^{5} - \frac{41}{119} a^{4} - \frac{113}{119} a^{3} + \frac{390}{119} a^{2} + \frac{1408}{119} a + \frac{877}{119} \),  \( \frac{281}{119} a^{5} - \frac{1069}{119} a^{4} - \frac{2508}{119} a^{3} + \frac{9530}{119} a^{2} + \frac{31565}{119} a + \frac{19450}{119} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 34.3096029889 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 6T9):

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

\(\Q(\sqrt{-3}) \)

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: 3.1.211.1 $\times$ 3.3.5697.1
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 ${\href{/LocalNumberField/2.6.0.1}{6} }$ R ${\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$3.6.7.5$x^{6} + 6 x^{2} + 3$$6$$1$$7$$D_{6}$$[3/2]_{2}^{2}$
211Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.211.2t1.1c1$1$ $ 211 $ $x^{2} - x + 53$ $C_2$ (as 2T1) $1$ $-1$
1.3_211.2t1.1c1$1$ $ 3 \cdot 211 $ $x^{2} - x - 158$ $C_2$ (as 2T1) $1$ $1$
2.3e3_211.6t3.1c1$2$ $ 3^{3} \cdot 211 $ $x^{6} - 17 x^{3} + 125$ $D_{6}$ (as 6T3) $1$ $-2$
2.3e3_211.3t2.1c1$2$ $ 3^{3} \cdot 211 $ $x^{3} - 15 x - 17$ $S_3$ (as 3T2) $1$ $2$
2.211.3t2.1c1$2$ $ 211 $ $x^{3} - 2 x - 3$ $S_3$ (as 3T2) $1$ $0$
2.3e2_211.6t3.1c1$2$ $ 3^{2} \cdot 211 $ $x^{6} + 2 x^{4} - 6 x^{3} + 4 x^{2} - 6 x + 9$ $D_{6}$ (as 6T3) $1$ $0$
* 4.3e6_211e2.6t9.1c1$4$ $ 3^{6} \cdot 211^{2}$ $x^{6} - 3 x^{5} - 12 x^{4} + 27 x^{3} + 141 x^{2} + 162 x + 57$ $S_3^2$ (as 6T9) $1$ $0$

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 *.