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Properties

Label 6.0.11374951703.2
Degree $6$
Signature $[0, 3]$
Discriminant $-\,47^{3}\cdot 331^{2}$
Root discriminant $47.42$
Ramified primes $47, 331$
Class number $8$
Class group $[8]$
Galois group $S_4\times C_2$ (as 6T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16753, -422, 1221, -123, 45, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 + 45*x^4 - 123*x^3 + 1221*x^2 - 422*x + 16753)
 
gp: K = bnfinit(x^6 - 2*x^5 + 45*x^4 - 123*x^3 + 1221*x^2 - 422*x + 16753, 1)
 

Normalized defining polynomial

\( x^{6} - 2 x^{5} + 45 x^{4} - 123 x^{3} + 1221 x^{2} - 422 x + 16753 \)

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:  \(-11374951703=-\,47^{3}\cdot 331^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.42$
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:  $47, 331$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2594379} a^{5} + \frac{37230}{864793} a^{4} + \frac{368431}{864793} a^{3} - \frac{379694}{864793} a^{2} - \frac{197914}{864793} a + \frac{1284112}{2594379}$

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

Class group and class number

$C_{8}$, which has order $8$

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:  \( \frac{2075}{864793} a^{5} - \frac{7774}{864793} a^{4} + \frac{51939}{864793} a^{3} - \frac{115881}{864793} a^{2} + \frac{1180168}{864793} a + \frac{969960}{864793} \),  \( \frac{15532}{864793} a^{5} - \frac{5678}{864793} a^{4} + \frac{405033}{864793} a^{3} - \frac{2880809}{864793} a^{2} - \frac{2442571}{864793} a - \frac{51780955}{864793} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 32.0451805365 \)
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

3.1.331.1

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

Sibling algebras

Twin sextic algebra: 4.2.331.1 $\times$ \(\Q(\sqrt{15557}) \)
Degree 6 sibling: 6.2.3765109013693.1
Degree 8 siblings: Deg 8, Deg 8
Degree 12 siblings: Deg 12, Deg 12, Deg 12, Deg 12, Deg 12, 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 ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }$ R ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$47$47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
331Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.47.2t1.1c1$1$ $ 47 $ $x^{2} - x + 12$ $C_2$ (as 2T1) $1$ $-1$
1.331.2t1.1c1$1$ $ 331 $ $x^{2} - x + 83$ $C_2$ (as 2T1) $1$ $-1$
1.47_331.2t1.1c1$1$ $ 47 \cdot 331 $ $x^{2} - x - 3889$ $C_2$ (as 2T1) $1$ $1$
2.47e2_331.6t3.2c1$2$ $ 47^{2} \cdot 331 $ $x^{6} - 2 x^{5} - 185 x^{4} + 574 x^{3} + 8261 x^{2} - 36084 x - 522416$ $D_{6}$ (as 6T3) $1$ $0$
* 2.331.3t2.1c1$2$ $ 331 $ $x^{3} - x^{2} + 3 x - 4$ $S_3$ (as 3T2) $1$ $0$
3.331.4t5.1c1$3$ $ 331 $ $x^{4} - x^{3} + x^{2} + x - 1$ $S_4$ (as 4T5) $1$ $1$
3.47e3_331e2.6t11.2c1$3$ $ 47^{3} \cdot 331^{2}$ $x^{6} - 2 x^{5} + 45 x^{4} - 123 x^{3} + 1221 x^{2} - 422 x + 16753$ $S_4\times C_2$ (as 6T11) $1$ $1$
* 3.47e3_331.6t11.2c1$3$ $ 47^{3} \cdot 331 $ $x^{6} - 2 x^{5} + 45 x^{4} - 123 x^{3} + 1221 x^{2} - 422 x + 16753$ $S_4\times C_2$ (as 6T11) $1$ $-1$
3.331e2.6t8.3c1$3$ $ 331^{2}$ $x^{4} - x^{3} + x^{2} + x - 1$ $S_4$ (as 4T5) $1$ $-1$

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 *.