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Properties

Label 6.2.78755933.3
Degree $6$
Signature $[2, 2]$
Discriminant $23^{2}\cdot 53^{3}$
Root discriminant $20.70$
Ramified primes $23, 53$
Class number $4$
Class group $[4]$
Galois group $D_{6}$ (as 6T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2183, -248, 521, 25, -40, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 40*x^4 + 25*x^3 + 521*x^2 - 248*x - 2183)
 
gp: K = bnfinit(x^6 - x^5 - 40*x^4 + 25*x^3 + 521*x^2 - 248*x - 2183, 1)
 

Normalized defining polynomial

\( x^{6} - x^{5} - 40 x^{4} + 25 x^{3} + 521 x^{2} - 248 x - 2183 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $6$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(78755933=23^{2}\cdot 53^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.70$
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:  $23, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{143335} a^{5} + \frac{13154}{28667} a^{4} + \frac{10333}{28667} a^{3} + \frac{3179}{28667} a^{2} - \frac{54924}{143335} a + \frac{65353}{143335}$

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:  $3$
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:  \( \frac{122}{143335} a^{5} - \frac{564}{28667} a^{4} - \frac{722}{28667} a^{3} + \frac{15167}{28667} a^{2} + \frac{36017}{143335} a - \frac{483699}{143335} \),  \( \frac{316}{143335} a^{5} - \frac{51}{28667} a^{4} - \frac{2810}{28667} a^{3} + \frac{1219}{28667} a^{2} + \frac{274221}{143335} a + \frac{441313}{143335} \),  \( \frac{4522}{143335} a^{5} - \frac{1637}{28667} a^{4} - \frac{30051}{28667} a^{3} + \frac{41938}{28667} a^{2} + \frac{1323242}{143335} a - \frac{1893859}{143335} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 9.85671118646 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_6$ (as 6T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12
The 6 conjugacy class representatives for $D_{6}$
Character table for $D_{6}$

Intermediate fields

\(\Q(\sqrt{53}) \), 3.1.23.1

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

Sibling algebras

Galois closure: Deg 12
Twin sextic algebra: data not computed
Degree 6 sibling: 6.0.1811386459.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.23.2t1.1c1$1$ $ 23 $ $x^{2} - x + 6$ $C_2$ (as 2T1) $1$ $-1$
1.23_53.2t1.1c1$1$ $ 23 \cdot 53 $ $x^{2} - x + 305$ $C_2$ (as 2T1) $1$ $-1$
* 1.53.2t1.1c1$1$ $ 53 $ $x^{2} - x - 13$ $C_2$ (as 2T1) $1$ $1$
* 2.23.3t2.1c1$2$ $ 23 $ $x^{3} - x^{2} + 1$ $S_3$ (as 3T2) $1$ $0$
* 2.23_53e2.6t3.1c1$2$ $ 23 \cdot 53^{2}$ $x^{6} - x^{5} - 40 x^{4} + 25 x^{3} + 521 x^{2} - 248 x - 2183$ $D_{6}$ (as 6T3) $1$ $0$

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 *.