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Properties

Label 8.0.314703872.4
Degree $8$
Signature $[0, 4]$
Discriminant $2^{17}\cdot 7^{4}$
Root discriminant $11.54$
Ramified primes $2, 7$
Class number $1$
Class group Trivial
Galois group $C_4\wr C_2$ (as 8T17)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, 0, 12, 0, 6, 0, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 2*x^6 + 6*x^4 + 12*x^2 + 8)
 
gp: K = bnfinit(x^8 + 2*x^6 + 6*x^4 + 12*x^2 + 8, 1)
 

Normalized defining polynomial

\( x^{8} + 2 x^{6} + 6 x^{4} + 12 x^{2} + 8 \)

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:  \(314703872=2^{17}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.54$
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:  $2, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{4} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3}$

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:  \( \frac{1}{4} a^{6} + \frac{3}{2} a^{2} + 1 \),  \( \frac{1}{4} a^{7} + \frac{3}{2} a^{3} + a^{2} + 2 a + 1 \),  \( \frac{1}{4} a^{7} + a^{6} + a^{4} + \frac{3}{2} a^{3} + 5 a^{2} + 7 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 13.4897264996 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 8T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.392.1

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

Sibling fields

Galois closure: data not computed
Degree 8 sibling: data not computed
Degree 16 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 ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ R ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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$2.4.11.9$x^{4} + 12 x^{2} + 10$$4$$1$$11$$C_4$$[3, 4]$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e3_7.2t1.2c1$1$ $ 2^{3} \cdot 7 $ $x^{2} + 14$ $C_2$ (as 2T1) $1$ $-1$
1.2e3.2t1.1c1$1$ $ 2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
* 1.7.2t1.1c1$1$ $ 7 $ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
1.2e4_7.4t1.2c1$1$ $ 2^{4} \cdot 7 $ $x^{4} + 28 x^{2} + 98$ $C_4$ (as 4T1) $0$ $-1$
1.2e4.4t1.1c1$1$ $ 2^{4}$ $x^{4} - 4 x^{2} + 2$ $C_4$ (as 4T1) $0$ $1$
1.2e4_7.4t1.2c2$1$ $ 2^{4} \cdot 7 $ $x^{4} + 28 x^{2} + 98$ $C_4$ (as 4T1) $0$ $-1$
1.2e4.4t1.1c2$1$ $ 2^{4}$ $x^{4} - 4 x^{2} + 2$ $C_4$ (as 4T1) $0$ $1$
2.2e8_7.4t3.7c1$2$ $ 2^{8} \cdot 7 $ $x^{4} - 4 x^{2} + 18$ $D_{4}$ (as 4T3) $1$ $0$
* 2.2e3_7.4t3.2c1$2$ $ 2^{3} \cdot 7 $ $x^{4} - x^{3} + x + 1$ $D_{4}$ (as 4T3) $1$ $0$
* 2.2e7_7.8t17.4c1$2$ $ 2^{7} \cdot 7 $ $x^{8} + 2 x^{6} + 6 x^{4} + 12 x^{2} + 8$ $C_4\wr C_2$ (as 8T17) $0$ $0$
* 2.2e7_7.8t17.4c2$2$ $ 2^{7} \cdot 7 $ $x^{8} + 2 x^{6} + 6 x^{4} + 12 x^{2} + 8$ $C_4\wr C_2$ (as 8T17) $0$ $0$
2.2e6_7.8t17.2c1$2$ $ 2^{6} \cdot 7 $ $x^{8} + 2 x^{6} + 6 x^{4} + 12 x^{2} + 8$ $C_4\wr C_2$ (as 8T17) $0$ $0$
2.2e6_7.8t17.2c2$2$ $ 2^{6} \cdot 7 $ $x^{8} + 2 x^{6} + 6 x^{4} + 12 x^{2} + 8$ $C_4\wr C_2$ (as 8T17) $0$ $0$

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 *.