Properties

Label 6.2.241856.1
Degree $6$
Signature $[2, 2]$
Discriminant $241856$
Root discriminant $7.89$
Ramified primes $2, 3779$
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 + 2*x^4 - 4*x^3 + 5*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^6 - 2*x^5 + 2*x^4 - 4*x^3 + 5*x^2 - 4*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 5, -4, 2, -2, 1]);
 

\(x^{6} - 2 x^{5} + 2 x^{4} - 4 x^{3} + 5 x^{2} - 4 x + 1\)  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:  \(241856\)\(\medspace = 2^{6}\cdot 3779\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $7.89$
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:  $2, 3779$
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:  \( a^{5} - a^{4} + a^{3} - 3 a^{2} + 2 a - 2 \),  \( a \),  \( a^{5} - 2 a^{4} + 2 a^{3} - 4 a^{2} + 4 a - 3 \)  Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1.85505134143 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{2}\cdot 1.85505134143 \cdot 1}{2\sqrt{241856}}\approx 0.297829171461$

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: 6.2.13815628323584.4
Degree 6 sibling: 6.2.13815628323584.4
Degree 10 sibling: 10.2.221050053177344.1
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 ${\href{/padicField/3.6.0.1}{6} }$ ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.6.0.1}{6} }$ ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.6.0.1}{6} }$ ${\href{/padicField/41.6.0.1}{6} }$ ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.6.0.1}{6} }$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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$$[2, 2, 2]^{3}$
$3779$Deg $2$$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$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.15116.2t1.a.a$1$ $ 2^{2} \cdot 3779 $ \(\Q(\sqrt{3779}) \) $C_2$ (as 2T1) $1$ $1$
* 5.241856.6t16.a.a$5$ $ 2^{6} \cdot 3779 $ 6.2.241856.1 $S_6$ (as 6T16) $1$ $1$
5.3655895296.12t183.a.a$5$ $ 2^{8} \cdot 3779^{2}$ 6.2.241856.1 $S_6$ (as 6T16) $1$ $1$
5.208...744.12t183.a.a$5$ $ 2^{10} \cdot 3779^{4}$ 6.2.241856.1 $S_6$ (as 6T16) $1$ $1$
5.138...584.6t16.a.a$5$ $ 2^{8} \cdot 3779^{3}$ 6.2.241856.1 $S_6$ (as 6T16) $1$ $1$
9.221...344.10t32.a.a$9$ $ 2^{12} \cdot 3779^{3}$ 6.2.241856.1 $S_6$ (as 6T16) $1$ $1$
9.763...224.20t145.a.a$9$ $ 2^{18} \cdot 3779^{6}$ 6.2.241856.1 $S_6$ (as 6T16) $1$ $1$
10.305...896.30t164.a.a$10$ $ 2^{20} \cdot 3779^{6}$ 6.2.241856.1 $S_6$ (as 6T16) $1$ $-2$
10.534...464.30t164.a.a$10$ $ 2^{18} \cdot 3779^{4}$ 6.2.241856.1 $S_6$ (as 6T16) $1$ $-2$
16.111...216.36t1252.a.a$16$ $ 2^{28} \cdot 3779^{8}$ 6.2.241856.1 $S_6$ (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 *.