↑

Properties

Label 6.2.5030912000.1
Degree $6$
Signature $[2, 2]$
Discriminant $2^{13}\cdot 5^{3}\cdot 17^{3}$
Root discriminant $41.39$
Ramified primes $2, 5, 17$
Class number $4$
Class group $[2, 2]$
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![-680, 0, -76, 0, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 + 6*x^4 - 76*x^2 - 680)
 
gp: K = bnfinit(x^6 + 6*x^4 - 76*x^2 - 680, 1)
 

Normalized defining polynomial

\( x^{6} + 6 x^{4} - 76 x^{2} - 680 \)

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:  \(5030912000=2^{13}\cdot 5^{3}\cdot 17^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.39$
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:  $2, 5, 17$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{40} a^{4} + \frac{1}{5} a^{2} - \frac{1}{2}$, $\frac{1}{40} a^{5} - \frac{1}{20} a^{3}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  \( \frac{1}{40} a^{5} + \frac{1}{5} a^{4} - \frac{3}{10} a^{3} + \frac{1}{10} a^{2} + \frac{1}{2} a - 18 \),  \( \frac{1}{40} a^{5} + \frac{1}{40} a^{4} - \frac{1}{20} a^{3} + \frac{1}{5} a^{2} - 2 a - \frac{7}{2} \),  \( \frac{1}{40} a^{5} - \frac{1}{40} a^{4} - \frac{1}{20} a^{3} - \frac{1}{5} a^{2} - 2 a + \frac{7}{2} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 262.007272193 \)
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

3.1.680.1

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

Sibling algebras

Twin sextic algebra: 4.2.43520.1 $\times$ \(\Q(\sqrt{-1}) \)
Degree 6 sibling: 6.0.7398400.2
Degree 8 siblings: 8.4.218945290240000.6, 8.0.1893990400.14
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 R ${\href{/LocalNumberField/3.6.0.1}{6} }$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }$

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
$2$2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.4.10.6$x^{4} + 6 x^{2} + 3$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e2.2t1.1c1$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
1.2e3_5_17.2t1.2c1$1$ $ 2^{3} \cdot 5 \cdot 17 $ $x^{2} + 170$ $C_2$ (as 2T1) $1$ $-1$
1.2e3_5_17.2t1.1c1$1$ $ 2^{3} \cdot 5 \cdot 17 $ $x^{2} - 170$ $C_2$ (as 2T1) $1$ $1$
2.2e5_5_17.6t3.1c1$2$ $ 2^{5} \cdot 5 \cdot 17 $ $x^{6} - 2 x^{5} + 2 x^{4} - 8 x^{3} + 25 x^{2} - 10 x + 2$ $D_{6}$ (as 6T3) $1$ $0$
* 2.2e3_5_17.3t2.1c1$2$ $ 2^{3} \cdot 5 \cdot 17 $ $x^{3} - x^{2} - 6 x + 10$ $S_3$ (as 3T2) $1$ $0$
3.2e9_5_17.4t5.1c1$3$ $ 2^{9} \cdot 5 \cdot 17 $ $x^{4} - 10 x^{2} - 8 x - 2$ $S_4$ (as 4T5) $1$ $1$
* 3.2e10_5e2_17e2.6t11.1c1$3$ $ 2^{10} \cdot 5^{2} \cdot 17^{2}$ $x^{6} + 6 x^{4} - 76 x^{2} - 680$ $S_4\times C_2$ (as 6T11) $1$ $1$
3.2e7_5_17.6t11.1c1$3$ $ 2^{7} \cdot 5 \cdot 17 $ $x^{6} + 6 x^{4} - 76 x^{2} - 680$ $S_4\times C_2$ (as 6T11) $1$ $-1$
3.2e10_5e2_17e2.6t8.3c1$3$ $ 2^{10} \cdot 5^{2} \cdot 17^{2}$ $x^{4} - 10 x^{2} - 8 x - 2$ $S_4$ (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 *.