Properties

Label 8.0.843308032.1
Degree $8$
Signature $[0, 4]$
Discriminant $2^{10}\cdot 7^{7}$
Root discriminant $13.05$
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![9, -18, 7, 14, -14, 0, 7, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 7*x^6 - 14*x^4 + 14*x^3 + 7*x^2 - 18*x + 9)
gp: K = bnfinit(x^8 - 4*x^7 + 7*x^6 - 14*x^4 + 14*x^3 + 7*x^2 - 18*x + 9, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut 4 x^{7} \) \(\mathstrut +\mathstrut 7 x^{6} \) \(\mathstrut -\mathstrut 14 x^{4} \) \(\mathstrut +\mathstrut 14 x^{3} \) \(\mathstrut +\mathstrut 7 x^{2} \) \(\mathstrut -\mathstrut 18 x \) \(\mathstrut +\mathstrut 9 \)

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:  \(843308032=2^{10}\cdot 7^{7}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $13.05$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{2}{9} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a$

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:  $3$
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:  \( \frac{7}{9} a^{7} - \frac{20}{9} a^{6} + \frac{8}{3} a^{5} + \frac{11}{3} a^{4} - \frac{68}{9} a^{3} + \frac{10}{9} a^{2} + \frac{25}{3} a - 4 \),  \( \frac{1}{9} a^{7} - \frac{5}{9} a^{6} + a^{5} - \frac{1}{3} a^{4} - \frac{11}{9} a^{3} + \frac{7}{9} a^{2} + \frac{2}{3} a - 1 \),  \( \frac{5}{9} a^{7} - \frac{10}{9} a^{6} + \frac{2}{3} a^{5} + \frac{10}{3} a^{4} - \frac{19}{9} a^{3} - \frac{22}{9} a^{2} + 3 a \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 33.7643922139 \)
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

\(\Q(\sqrt{-7}) \), 4.0.1372.1

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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.4.6.9$x^{4} + 2 x^{3} + 6$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
$7$7.8.7.1$x^{8} + 14$$8$$1$$7$$D_{8}$$[\ ]_{8}^{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_7.2t1.1c1$1$ $ 2^{2} \cdot 7 $ $x^{2} - 7$ $C_2$ (as 2T1) $1$ $1$
* 1.7.2t1.1c1$1$ $ 7 $ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
1.2e2.2t1.1c1$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
* 2.2e2_7e2.4t3.1c1$2$ $ 2^{2} \cdot 7^{2}$ $x^{4} - 2 x^{3} + 5 x^{2} - 4 x - 3$ $D_{4}$ (as 4T3) $1$ $0$
* 2.2e4_7e2.8t6.2c1$2$ $ 2^{4} \cdot 7^{2}$ $x^{8} - 4 x^{7} + 7 x^{6} - 14 x^{4} + 14 x^{3} + 7 x^{2} - 18 x + 9$ $D_{8}$ (as 8T6) $1$ $0$
* 2.2e4_7e2.8t6.2c2$2$ $ 2^{4} \cdot 7^{2}$ $x^{8} - 4 x^{7} + 7 x^{6} - 14 x^{4} + 14 x^{3} + 7 x^{2} - 18 x + 9$ $D_{8}$ (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 *.