Properties

Label 6.2.294592.1
Degree $6$
Signature $[2, 2]$
Discriminant $294592$
Root discriminant $8.16$
Ramified primes $2, 4603$
Class number $1$
Class group trivial
Galois group $S_6$ (as 6T16)

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Show commands: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 + 4*x^4 - 4*x^3 + x^2 - 1)
 
gp: K = bnfinit(x^6 - 2*x^5 + 4*x^4 - 4*x^3 + x^2 - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 1, -4, 4, -2, 1]);
 

\(x^{6} - 2 x^{5} + 4 x^{4} - 4 x^{3} + x^{2} - 1\)  Toggle raw display

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:  \(294592\)\(\medspace = 2^{6}\cdot 4603\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $8.16$
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:  $2, 4603$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$  Toggle raw display

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 $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( a^{5} - 2 a^{4} + 4 a^{3} - 4 a^{2} + a \),  \( a^{2} - a + 1 \),  \( a - 1 \)  Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 2.41068187178 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{2}\cdot 2.41068187178 \cdot 1}{2\sqrt{294592}}\approx 0.350686590701$

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: 6.2.24966800442112.1
Degree 6 sibling: 6.2.24966800442112.1
Degree 10 sibling: 10.2.399468807073792.1
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: Deg 30, Deg 30, Deg 30, Deg 30, Deg 30, Deg 30
Degree 36 sibling: Deg 36
Degree 40 siblings: Deg 40, Deg 40, Deg 40
Degree 45 sibling: Deg 45

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ ${\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.6.0.1}{6} }$ ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.6.0.1}{6} }$ ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.6.0.1}{6} }$ ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
$2$2.6.6.6$x^{6} - 13 x^{4} + 7 x^{2} - 3$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
$4603$Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.18412.2t1.a.a$1$ $ 2^{2} \cdot 4603 $ \(\Q(\sqrt{4603}) \) $C_2$ (as 2T1) $1$ $1$
* 5.294592.6t16.a.a$5$ $ 2^{6} \cdot 4603 $ 6.2.294592.1 $S_6$ (as 6T16) $1$ $1$
5.5424027904.12t183.a.a$5$ $ 2^{8} \cdot 4603^{2}$ 6.2.294592.1 $S_6$ (as 6T16) $1$ $1$
5.459...144.12t183.a.a$5$ $ 2^{10} \cdot 4603^{4}$ 6.2.294592.1 $S_6$ (as 6T16) $1$ $1$
5.249...112.6t16.a.a$5$ $ 2^{8} \cdot 4603^{3}$ 6.2.294592.1 $S_6$ (as 6T16) $1$ $1$
9.399...792.10t32.a.a$9$ $ 2^{12} \cdot 4603^{3}$ 6.2.294592.1 $S_6$ (as 6T16) $1$ $1$
9.249...176.20t145.a.a$9$ $ 2^{18} \cdot 4603^{6}$ 6.2.294592.1 $S_6$ (as 6T16) $1$ $1$
10.997...704.30t164.a.a$10$ $ 2^{20} \cdot 4603^{6}$ 6.2.294592.1 $S_6$ (as 6T16) $1$ $-2$
10.117...864.30t164.a.a$10$ $ 2^{18} \cdot 4603^{4}$ 6.2.294592.1 $S_6$ (as 6T16) $1$ $-2$
16.540...416.36t1252.a.a$16$ $ 2^{28} \cdot 4603^{8}$ 6.2.294592.1 $S_6$ (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 *.