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Properties

Label 6.0.202365391311.1
Degree $6$
Signature $[0, 3]$
Discriminant $-\,3^{3}\cdot 19^{3}\cdot 103^{3}$
Root discriminant $76.62$
Ramified primes $3, 19, 103$
Class number $64$
Class group $[2, 2, 4, 4]$
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![1027, -822, 888, -133, 69, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 + 69*x^4 - 133*x^3 + 888*x^2 - 822*x + 1027)
 
gp: K = bnfinit(x^6 - 3*x^5 + 69*x^4 - 133*x^3 + 888*x^2 - 822*x + 1027, 1)
 

Normalized defining polynomial

\( x^{6} - 3 x^{5} + 69 x^{4} - 133 x^{3} + 888 x^{2} - 822 x + 1027 \)

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:  \(-202365391311=-\,3^{3}\cdot 19^{3}\cdot 103^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.62$
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:  $3, 19, 103$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{171} a^{4} - \frac{2}{171} a^{3} + \frac{3}{19} a^{2} - \frac{26}{171} a - \frac{47}{171}$, $\frac{1}{171} a^{5} + \frac{23}{171} a^{3} + \frac{28}{171} a^{2} + \frac{8}{19} a + \frac{77}{171}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{4}$, which has order $64$

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:  \( \frac{1}{171} a^{4} - \frac{2}{171} a^{3} + \frac{3}{19} a^{2} - \frac{26}{171} a + \frac{124}{171} \),  \( \frac{2}{171} a^{4} - \frac{4}{171} a^{3} + \frac{37}{57} a^{2} - \frac{109}{171} a + \frac{134}{171} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 18.2058029492 \)
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

3.3.1957.1

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

Sibling algebras

Twin sextic algebra: 4.4.1957.1 $\times$ \(\Q(\sqrt{-3}) \)
Degree 6 sibling: 6.0.103405923.1
Degree 8 siblings: 8.0.310217769.2, 8.0.1188087212386881.9
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 ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
103Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.19_103.2t1.1c1$1$ $ 19 \cdot 103 $ $x^{2} - x - 489$ $C_2$ (as 2T1) $1$ $1$
1.3_19_103.2t1.1c1$1$ $ 3 \cdot 19 \cdot 103 $ $x^{2} - x + 1468$ $C_2$ (as 2T1) $1$ $-1$
2.3e2_19_103.6t3.4c1$2$ $ 3^{2} \cdot 19 \cdot 103 $ $x^{6} - x^{5} + 10 x^{4} - 11 x^{3} + 91 x^{2} - 90 x + 100$ $D_{6}$ (as 6T3) $1$ $-2$
* 2.19_103.3t2.1c1$2$ $ 19 \cdot 103 $ $x^{3} - x^{2} - 9 x + 10$ $S_3$ (as 3T2) $1$ $2$
3.19_103.4t5.1c1$3$ $ 19 \cdot 103 $ $x^{4} - 4 x^{2} - x + 1$ $S_4$ (as 4T5) $1$ $3$
* 3.3e3_19e2_103e2.6t11.2c1$3$ $ 3^{3} \cdot 19^{2} \cdot 103^{2}$ $x^{6} - 3 x^{5} + 69 x^{4} - 133 x^{3} + 888 x^{2} - 822 x + 1027$ $S_4\times C_2$ (as 6T11) $1$ $-3$
3.3e3_19_103.6t11.2c1$3$ $ 3^{3} \cdot 19 \cdot 103 $ $x^{6} - 3 x^{5} + 69 x^{4} - 133 x^{3} + 888 x^{2} - 822 x + 1027$ $S_4\times C_2$ (as 6T11) $1$ $-3$
3.19e2_103e2.6t8.1c1$3$ $ 19^{2} \cdot 103^{2}$ $x^{4} - 4 x^{2} - x + 1$ $S_4$ (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 *.