Properties

Label 8.2.1438646272.4
Degree $8$
Signature $[2, 3]$
Discriminant $-\,2^{22}\cdot 7^{3}$
Root discriminant $13.96$
Ramified primes $2, 7$
Class number $1$
Class group Trivial
Galois Group $D_{8}$ (as 8T6)

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Show commands for: Magma / SageMath / Pari/GP

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

Normalized defining polynomial

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

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

Invariants

Degree:  $8$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[2, 3]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-1438646272=-\,2^{22}\cdot 7^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $13.96$
magma: Abs(Discriminant(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(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 \(a\))

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

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

Class group and class number

Trivial Abelian group, 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:  $4$
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:  \( \frac{1}{7} a^{7} - \frac{5}{7} a^{6} + \frac{8}{7} a^{5} - \frac{13}{7} a^{4} + \frac{24}{7} a^{3} - 3 a^{2} + \frac{16}{7} a - \frac{17}{7} \),  \( \frac{2}{7} a^{7} - \frac{9}{7} a^{6} + \frac{12}{7} a^{5} - \frac{19}{7} a^{4} + \frac{30}{7} a^{3} - \frac{22}{7} a^{2} + \frac{12}{7} a - \frac{13}{7} \),  \( \frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{3}{7} a^{4} - \frac{11}{7} a^{3} + \frac{2}{7} a^{2} - a + \frac{1}{7} \),  \( \frac{4}{7} a^{7} - \frac{16}{7} a^{6} + \frac{18}{7} a^{5} - \frac{39}{7} a^{4} + \frac{59}{7} a^{3} - \frac{40}{7} a^{2} + \frac{31}{7} a - \frac{31}{7} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 55.5269775606 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_8$ (as 8T6):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.2.1792.1

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

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 ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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$2.8.22.83$x^{8} + 4 x^{7} + 2 x^{4} + 4 x^{2} + 14$$8$$1$$22$$D_4$$[2, 3, 7/2]$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e3.2t1.1c1$1$ $ 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) $1$ $-1$
1.2e3_7.2t1.2c1$1$ $ 2^{3} \cdot 7 $ $x^{2} + 14$ $C_2$ (as 2T1) $1$ $-1$
* 2.2e5_7.4t3.1c1$2$ $ 2^{5} \cdot 7 $ $x^{4} - 2 x^{2} - 4 x - 2$ $D_{4}$ (as 4T3) $1$ $0$
* 2.2e7_7.8t6.1c1$2$ $ 2^{7} \cdot 7 $ $x^{8} - 4 x^{7} + 4 x^{6} - 8 x^{5} + 14 x^{4} - 8 x^{3} + 4 x^{2} - 8 x - 2$ $D_{8}$ (as 8T6) $1$ $0$
* 2.2e7_7.8t6.1c2$2$ $ 2^{7} \cdot 7 $ $x^{8} - 4 x^{7} + 4 x^{6} - 8 x^{5} + 14 x^{4} - 8 x^{3} + 4 x^{2} - 8 x - 2$ $D_{8}$ (as 8T6) $1$ $0$

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 *.