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Properties

Label 8.0.49401285696.3
Degree $8$
Signature $[0, 4]$
Discriminant $2^{6}\cdot 3^{8}\cdot 7^{6}$
Root discriminant $21.71$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $V_4^2:S_3$ (as 8T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, -7, -7, 14, 4, -11, 5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + 5*x^6 - 11*x^5 + 4*x^4 + 14*x^3 - 7*x^2 - 7*x + 7)
 
gp: K = bnfinit(x^8 - 2*x^7 + 5*x^6 - 11*x^5 + 4*x^4 + 14*x^3 - 7*x^2 - 7*x + 7, 1)
 

Normalized defining polynomial

\( x^{8} - 2 x^{7} + 5 x^{6} - 11 x^{5} + 4 x^{4} + 14 x^{3} - 7 x^{2} - 7 x + 7 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(49401285696=2^{6}\cdot 3^{8}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.71$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{78} a^{7} - \frac{11}{78} a^{6} - \frac{11}{78} a^{4} + \frac{25}{78} a^{3} - \frac{1}{26} a^{2} + \frac{10}{39} a - \frac{31}{78}$

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:  $3$
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:  \( \frac{5}{39} a^{7} - \frac{1}{13} a^{6} + \frac{23}{39} a^{4} - \frac{58}{13} a^{3} + \frac{125}{13} a^{2} - \frac{329}{39} a + \frac{48}{13} \),  \( \frac{2}{39} a^{7} - \frac{3}{13} a^{6} - \frac{61}{39} a^{4} - \frac{5}{13} a^{3} + \frac{11}{13} a^{2} + \frac{1}{39} a - \frac{12}{13} \),  \( \frac{29}{39} a^{7} - \frac{46}{39} a^{6} + 3 a^{5} - \frac{241}{39} a^{4} - \frac{55}{39} a^{3} + \frac{140}{13} a^{2} + \frac{73}{39} a - \frac{236}{39} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 434.560571828 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:S_4$ (as 8T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 10 conjugacy class representatives for $V_4^2:S_3$
Character table for $V_4^2:S_3$

Intermediate fields

\(\Q(\sqrt{-7}) \)

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: 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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
$7$7.4.3.1$x^{4} + 14$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.4.3.1$x^{4} + 14$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.7.2t1.1c1$1$ $ 7 $ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
2.3e4_7.3t2.1c1$2$ $ 3^{4} \cdot 7 $ $x^{3} - 3 x - 5$ $S_3$ (as 3T2) $1$ $0$
3.2e6_3e4_7.4t5.1c1$3$ $ 2^{6} \cdot 3^{4} \cdot 7 $ $x^{4} - 2 x^{3} + 6 x^{2} + 2 x - 1$ $S_4$ (as 4T5) $1$ $1$
3.2e6_3e4_7e3.4t5.1c1$3$ $ 2^{6} \cdot 3^{4} \cdot 7^{3}$ $x^{4} - 56 x + 84$ $S_4$ (as 4T5) $1$ $1$
3.2e6_3e4_7e3.4t5.2c1$3$ $ 2^{6} \cdot 3^{4} \cdot 7^{3}$ $x^{4} - 2 x^{3} - 30 x^{2} - 32 x + 130$ $S_4$ (as 4T5) $1$ $1$
3.2e6_3e4_7e2.6t8.6c1$3$ $ 2^{6} \cdot 3^{4} \cdot 7^{2}$ $x^{4} - 2 x^{3} + 6 x^{2} + 2 x - 1$ $S_4$ (as 4T5) $1$ $-1$
3.2e6_3e4_7e2.6t8.9c1$3$ $ 2^{6} \cdot 3^{4} \cdot 7^{2}$ $x^{4} - 2 x^{3} - 30 x^{2} - 32 x + 130$ $S_4$ (as 4T5) $1$ $-1$
3.2e6_3e4_7e2.6t8.8c1$3$ $ 2^{6} \cdot 3^{4} \cdot 7^{2}$ $x^{4} - 56 x + 84$ $S_4$ (as 4T5) $1$ $-1$
* 6.2e6_3e8_7e5.8t34.1c1$6$ $ 2^{6} \cdot 3^{8} \cdot 7^{5}$ $x^{8} - 2 x^{7} + 5 x^{6} - 11 x^{5} + 4 x^{4} + 14 x^{3} - 7 x^{2} - 7 x + 7$ $V_4^2:S_3$ (as 8T34) $1$ $0$

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 *.