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Properties

Label 6.4.1056655611.1
Degree $6$
Signature $[4, 1]$
Discriminant $-\,3^{8}\cdot 11^{5}$
Root discriminant $31.91$
Ramified primes $3, 11$
Class number $1$
Class group Trivial
Galois group $S_6$ (as 6T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, -66, 99, -44, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 44*x^3 + 99*x^2 - 66*x + 11)
 
gp: K = bnfinit(x^6 - 44*x^3 + 99*x^2 - 66*x + 11, 1)
 

Normalized defining polynomial

\( x^{6} - 44 x^{3} + 99 x^{2} - 66 x + 11 \)

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

Invariants

Degree:  $6$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 1]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1056655611=-\,3^{8}\cdot 11^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.91$
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:  $3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{14} a^{5} + \frac{3}{14} a^{4} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a + \frac{1}{14}$

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:  $4$
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:  \( a^{5} + a^{4} + a^{3} - 43 a^{2} + 56 a - 10 \),  \( \frac{9}{14} a^{5} + \frac{13}{14} a^{4} + \frac{9}{7} a^{3} - \frac{185}{7} a^{2} + \frac{174}{7} a - \frac{47}{14} \),  \( \frac{4}{7} a^{5} + \frac{17}{14} a^{4} + \frac{37}{14} a^{3} - \frac{269}{14} a^{2} + \frac{115}{7} a - \frac{41}{14} \),  \( \frac{8}{7} a^{5} + \frac{13}{14} a^{4} + \frac{11}{14} a^{3} - \frac{685}{14} a^{2} + \frac{489}{7} a - \frac{201}{14} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 253.822851411 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_6$ (as 6T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 720
The 11 conjugacy class representatives for $S_6$
Character table for $S_6$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling algebras

Twin sextic algebra: 6.0.8732691.3
Degree 6 sibling: 6.0.8732691.3
Degree 10 sibling: 10.4.838858813116291.1
Degree 12 siblings: Deg 12, Deg 12
Degree 15 siblings: Deg 15, Deg 15
Degree 20 siblings: Deg 20, Deg 20, Deg 20
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/5.5.0.1}{5} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$3.6.8.1$x^{6} + 6 x^{5} + 18 x^{2} + 9$$3$$2$$8$$C_3^2:C_4$$[2, 2]^{4}$
$11$11.6.5.2$x^{6} + 33$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.11.2t1.1c1$1$ $ 11 $ $x^{2} - x + 3$ $C_2$ (as 2T1) $1$ $-1$
* 5.3e8_11e5.6t16.2c1$5$ $ 3^{8} \cdot 11^{5}$ $x^{6} - 44 x^{3} + 99 x^{2} - 66 x + 11$ $S_6$ (as 6T16) $1$ $3$
5.3e8_11e4.12t183.3c1$5$ $ 3^{8} \cdot 11^{4}$ $x^{6} - 44 x^{3} + 99 x^{2} - 66 x + 11$ $S_6$ (as 6T16) $1$ $1$
5.3e8_11e4.12t183.4c1$5$ $ 3^{8} \cdot 11^{4}$ $x^{6} - 44 x^{3} + 99 x^{2} - 66 x + 11$ $S_6$ (as 6T16) $1$ $-3$
5.3e8_11e3.6t16.2c1$5$ $ 3^{8} \cdot 11^{3}$ $x^{6} - 44 x^{3} + 99 x^{2} - 66 x + 11$ $S_6$ (as 6T16) $1$ $-1$
9.3e16_11e7.10t32.2c1$9$ $ 3^{16} \cdot 11^{7}$ $x^{6} - 44 x^{3} + 99 x^{2} - 66 x + 11$ $S_6$ (as 6T16) $1$ $3$
9.3e16_11e8.20t145.2c1$9$ $ 3^{16} \cdot 11^{8}$ $x^{6} - 44 x^{3} + 99 x^{2} - 66 x + 11$ $S_6$ (as 6T16) $1$ $-3$
10.3e16_11e8.30t176.3c1$10$ $ 3^{16} \cdot 11^{8}$ $x^{6} - 44 x^{3} + 99 x^{2} - 66 x + 11$ $S_6$ (as 6T16) $1$ $-2$
10.3e16_11e8.30t176.4c1$10$ $ 3^{16} \cdot 11^{8}$ $x^{6} - 44 x^{3} + 99 x^{2} - 66 x + 11$ $S_6$ (as 6T16) $1$ $2$
16.3e32_11e14.36t1252.2c1$16$ $ 3^{32} \cdot 11^{14}$ $x^{6} - 44 x^{3} + 99 x^{2} - 66 x + 11$ $S_6$ (as 6T16) $1$ $0$

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 *.