↑

Properties

Label 6.0.503792.1
Degree $6$
Signature $[0, 3]$
Discriminant $-\,2^{4}\cdot 23\cdot 37^{2}$
Root discriminant $8.92$
Ramified primes $2, 23, 37$
Class number $1$
Class group Trivial
Galois group $S_4\times C_2$ (as 6T11)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -8, 14, -13, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 + 9*x^4 - 13*x^3 + 14*x^2 - 8*x + 2)
 
gp: K = bnfinit(x^6 - 3*x^5 + 9*x^4 - 13*x^3 + 14*x^2 - 8*x + 2, 1)
 

Normalized defining polynomial

\( x^{6} - 3 x^{5} + 9 x^{4} - 13 x^{3} + 14 x^{2} - 8 x + 2 \)

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:  \(-503792=-\,2^{4}\cdot 23\cdot 37^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $8.92$
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:  $2, 23, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

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

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:  $2$
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:  \( a^{4} - 2 a^{3} + 6 a^{2} - 5 a + 3 \),  \( a^{4} - 2 a^{3} + 5 a^{2} - 4 a + 1 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6.64934608307 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_4$ (as 6T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 10 conjugacy class representatives for $S_4\times C_2$
Character table for $S_4\times C_2$

Intermediate fields

3.3.148.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling algebras

Twin sextic algebra: 4.4.78292.1 $\times$ \(\Q(\sqrt{-851}) \)
Degree 6 sibling: 6.0.18640304.1
Degree 8 siblings: 8.0.6129637264.1, 8.0.8391473414416.1
Degree 12 siblings: 12.0.4967752261334272.1, Deg 12, Deg 12, Deg 12, 12.0.347460933212416.2, Deg 12
Degree 16 sibling: Deg 16
Degree 24 siblings: 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$37$37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.23_37.2t1.1c1$1$ $ 23 \cdot 37 $ $x^{2} - x + 213$ $C_2$ (as 2T1) $1$ $-1$
1.37.2t1.1c1$1$ $ 37 $ $x^{2} - x - 9$ $C_2$ (as 2T1) $1$ $1$
1.23.2t1.1c1$1$ $ 23 $ $x^{2} - x + 6$ $C_2$ (as 2T1) $1$ $-1$
2.2e2_23e2_37.6t3.1c1$2$ $ 2^{2} \cdot 23^{2} \cdot 37 $ $x^{6} - 3 x^{5} + 88 x^{4} - 159 x^{3} + 1578 x^{2} - 2273 x + 9103$ $D_{6}$ (as 6T3) $1$ $-2$
* 2.2e2_37.3t2.1c1$2$ $ 2^{2} \cdot 37 $ $x^{3} - x^{2} - 3 x + 1$ $S_3$ (as 3T2) $1$ $2$
3.2e2_23e2_37.4t5.1c1$3$ $ 2^{2} \cdot 23^{2} \cdot 37 $ $x^{4} - x^{3} - 10 x^{2} + 8 x + 18$ $S_4$ (as 4T5) $1$ $3$
* 3.2e2_23_37.6t11.1c1$3$ $ 2^{2} \cdot 23 \cdot 37 $ $x^{6} - 3 x^{5} + 9 x^{4} - 13 x^{3} + 14 x^{2} - 8 x + 2$ $S_4\times C_2$ (as 6T11) $1$ $-3$
3.2e2_23_37e2.6t11.1c1$3$ $ 2^{2} \cdot 23 \cdot 37^{2}$ $x^{6} - 3 x^{5} + 9 x^{4} - 13 x^{3} + 14 x^{2} - 8 x + 2$ $S_4\times C_2$ (as 6T11) $1$ $-3$
3.2e2_23e2_37e2.6t8.1c1$3$ $ 2^{2} \cdot 23^{2} \cdot 37^{2}$ $x^{4} - x^{3} - 10 x^{2} + 8 x + 18$ $S_4$ (as 4T5) $1$ $3$

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 *.