Properties

Label 6.2.201933.1
Degree $6$
Signature $[2, 2]$
Discriminant $3^{6}\cdot 277$
Root discriminant $7.66$
Ramified primes $3, 277$
Class number $1$
Class group Trivial
Galois group $S_6$ (as 6T16)

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Show commands for: SageMath / Pari/GP / Magma

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

Normalized defining polynomial

\( x^{6} - 3 x^{4} + 3 x^{2} - 3 x + 1 \)

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

Invariants

Degree:  $6$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(201933=3^{6}\cdot 277\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $7.66$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 277$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $3$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( a \),  \( a^{5} - 3 a^{3} + 3 a - 2 \),  \( 2 a^{5} + a^{4} - 6 a^{3} - 3 a^{2} + 6 a - 2 \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 2.00432526068 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$S_6$ (as 6T16):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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: 6.2.15494117157.3
Degree 6 sibling: 6.2.15494117157.3
Degree 10 sibling: 10.2.11295211407453.1
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 ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.6.6.1$x^{6} + 3 x^{5} - 2$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
277Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.277.2t1.a.a$1$ $ 277 $ $x^{2} - x - 69$ $C_2$ (as 2T1) $1$ $1$
* 5.201933.6t16.a.a$5$ $ 3^{6} \cdot 277 $ $x^{6} - 3 x^{4} + 3 x^{2} - 3 x + 1$ $S_6$ (as 6T16) $1$ $1$
5.55935441.12t183.a.a$5$ $ 3^{6} \cdot 277^{2}$ $x^{6} - 3 x^{4} + 3 x^{2} - 3 x + 1$ $S_6$ (as 6T16) $1$ $1$
5.4291870452489.12t183.a.a$5$ $ 3^{6} \cdot 277^{4}$ $x^{6} - 3 x^{4} + 3 x^{2} - 3 x + 1$ $S_6$ (as 6T16) $1$ $1$
5.15494117157.6t16.a.a$5$ $ 3^{6} \cdot 277^{3}$ $x^{6} - 3 x^{4} + 3 x^{2} - 3 x + 1$ $S_6$ (as 6T16) $1$ $1$
9.11295211407453.10t32.a.a$9$ $ 3^{12} \cdot 277^{3}$ $x^{6} - 3 x^{4} + 3 x^{2} - 3 x + 1$ $S_6$ (as 6T16) $1$ $1$
9.240067666474841762649.20t145.a.a$9$ $ 3^{12} \cdot 277^{6}$ $x^{6} - 3 x^{4} + 3 x^{2} - 3 x + 1$ $S_6$ (as 6T16) $1$ $1$
10.2160608998273575863841.30t164.a.a$10$ $ 3^{14} \cdot 277^{6}$ $x^{6} - 3 x^{4} + 3 x^{2} - 3 x + 1$ $S_6$ (as 6T16) $1$ $-2$
10.28158962038780329.30t164.a.a$10$ $ 3^{14} \cdot 277^{4}$ $x^{6} - 3 x^{4} + 3 x^{2} - 3 x + 1$ $S_6$ (as 6T16) $1$ $-2$
16.9789223988907057071863085399361.36t1252.a.a$16$ $ 3^{24} \cdot 277^{8}$ $x^{6} - 3 x^{4} + 3 x^{2} - 3 x + 1$ $S_6$ (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 *.