↑

Properties

Label 6.0.548864.1
Degree $6$
Signature $[0, 3]$
Discriminant $-\,2^{13}\cdot 67$
Root discriminant $9.05$
Ramified primes $2, 67$
Class number $1$
Class group Trivial
Galois group $C_3^2:D_4$ (as 6T13)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

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

Normalized defining polynomial

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

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:  \(-548864=-\,2^{13}\cdot 67\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $9.05$
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:  $2, 67$
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 \(a\))

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

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:  \( a^{2} - a + 1 \),  \( a^{5} - a^{4} + a^{3} - 3 a^{2} + 2 a - 1 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6.07433603242 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\wr C_2$ (as 6T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $C_3^2:D_4$
Character table for $C_3^2:D_4$

Intermediate fields

\(\Q(\sqrt{-2}) \)

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

Sibling algebras

Twin sextic algebra: 6.4.615962624.1
Degree 6 sibling: 6.4.615962624.1
Degree 9 sibling: Deg 9
Degree 12 siblings: Deg 12, Deg 12, Deg 12, Deg 12, Deg 12, Deg 12
Degree 18 siblings: Deg 18, Deg 18, Deg 18
Degree 24 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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

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$2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.4.10.8$x^{4} + 2 x^{2} + 3$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
67Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e3_67.2t1.2c1$1$ $ 2^{3} \cdot 67 $ $x^{2} + 134$ $C_2$ (as 2T1) $1$ $-1$
* 1.2e3.2t1.2c1$1$ $ 2^{3}$ $x^{2} + 2$ $C_2$ (as 2T1) $1$ $-1$
1.2e2_67.2t1.1c1$1$ $ 2^{2} \cdot 67 $ $x^{2} - 67$ $C_2$ (as 2T1) $1$ $1$
2.2e7_67.4t3.2c1$2$ $ 2^{7} \cdot 67 $ $x^{4} + 14 x^{2} - 18$ $D_{4}$ (as 4T3) $1$ $0$
* 4.2e10_67.6t13.1c1$4$ $ 2^{10} \cdot 67 $ $x^{6} - 2 x^{5} + 3 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.2e12_67e3.12t36.1c1$4$ $ 2^{12} \cdot 67^{3}$ $x^{6} - 2 x^{5} + 3 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.2e9_67e2.6t13.1c1$4$ $ 2^{9} \cdot 67^{2}$ $x^{6} - 2 x^{5} + 3 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2$ $C_3^2:D_4$ (as 6T13) $1$ $2$
4.2e13_67e2.12t34.1c1$4$ $ 2^{13} \cdot 67^{2}$ $x^{6} - 2 x^{5} + 3 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2$ $C_3^2:D_4$ (as 6T13) $1$ $-2$

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 *.