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Properties

Label 6.0.3421872.4
Degree $6$
Signature $[0, 3]$
Discriminant $-\,2^{4}\cdot 3^{3}\cdot 89^{2}$
Root discriminant $12.28$
Ramified primes $2, 3, 89$
Class number $4$
Class group $[4]$
Galois group $\PGL(2,5)$ (as 6T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -10, 14, -12, 6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 + 6*x^4 - 12*x^3 + 14*x^2 - 10*x + 4)
 
gp: K = bnfinit(x^6 - 2*x^5 + 6*x^4 - 12*x^3 + 14*x^2 - 10*x + 4, 1)
 

Normalized defining polynomial

\( x^{6} - 2 x^{5} + 6 x^{4} - 12 x^{3} + 14 x^{2} - 10 x + 4 \)

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:  \(-3421872=-\,2^{4}\cdot 3^{3}\cdot 89^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.28$
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:  $2, 3, 89$
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 \(a\))

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

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( a^{5} - a^{4} + 5 a^{3} - 7 a^{2} + 7 a - 3 \),  \( a^{5} - 2 a^{4} + 5 a^{3} - 11 a^{2} + 10 a - 3 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 5.3725526246 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_5$ (as 6T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 7 conjugacy class representatives for $\PGL(2,5)$
Character table for $\PGL(2,5)$

Intermediate fields

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

Sibling algebras

Twin sextic algebra: \(\Q\) $\times$ 5.3.380208.1
Degree 5 sibling: 5.3.380208.1
Degree 10 siblings: 10.0.3903069328128.1, 10.4.433674369792.1
Degree 12 sibling: 12.0.11709207984384.1
Degree 15 sibling: 15.3.1483978183108890624.1
Degree 20 siblings: Deg 20, 20.0.15233950180173557331984384.1, Deg 20
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
4.2e4_3e3_89e2.10t12.2c1$4$ $ 2^{4} \cdot 3^{3} \cdot 89^{2}$ $x^{6} - 2 x^{5} + 6 x^{4} - 12 x^{3} + 14 x^{2} - 10 x + 4$ $\PGL(2,5)$ (as 6T14) $1$ $-2$
4.2e4_3_89e2.5t5.2c1$4$ $ 2^{4} \cdot 3 \cdot 89^{2}$ $x^{6} - 2 x^{5} + 6 x^{4} - 12 x^{3} + 14 x^{2} - 10 x + 4$ $\PGL(2,5)$ (as 6T14) $1$ $2$
5.2e4_3e2_89e2.10t13.2c1$5$ $ 2^{4} \cdot 3^{2} \cdot 89^{2}$ $x^{6} - 2 x^{5} + 6 x^{4} - 12 x^{3} + 14 x^{2} - 10 x + 4$ $\PGL(2,5)$ (as 6T14) $1$ $1$
* 5.2e4_3e3_89e2.6t14.2c1$5$ $ 2^{4} \cdot 3^{3} \cdot 89^{2}$ $x^{6} - 2 x^{5} + 6 x^{4} - 12 x^{3} + 14 x^{2} - 10 x + 4$ $\PGL(2,5)$ (as 6T14) $1$ $-1$
6.2e4_3e3_89e4.20t35.2c1$6$ $ 2^{4} \cdot 3^{3} \cdot 89^{4}$ $x^{6} - 2 x^{5} + 6 x^{4} - 12 x^{3} + 14 x^{2} - 10 x + 4$ $\PGL(2,5)$ (as 6T14) $1$ $0$

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 *.