Properties

Label 8.0.21381376.2
Degree $8$
Signature $[0, 4]$
Discriminant $2^{8}\cdot 17^{4}$
Root discriminant $8.25$
Ramified primes $2, 17$
Class number $1$
Class group Trivial
Galois Group $D_4$ (as 8T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 5, 0, 4, 0, 5, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 5*x^6 + 4*x^4 + 5*x^2 + 1)
gp: K = bnfinit(x^8 + 5*x^6 + 4*x^4 + 5*x^2 + 1, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut +\mathstrut 5 x^{6} \) \(\mathstrut +\mathstrut 4 x^{4} \) \(\mathstrut +\mathstrut 5 x^{2} \) \(\mathstrut +\mathstrut 1 \)

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:  \(21381376=2^{8}\cdot 17^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $8.25$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 17$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{8} a + \frac{1}{8}$

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:  \( \frac{3}{4} a^{7} + \frac{7}{2} a^{5} + \frac{3}{2} a^{3} + \frac{9}{4} a \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + a^{5} - a^{4} + \frac{5}{4} a - \frac{1}{4} \),  \( \frac{7}{8} a^{7} - \frac{1}{8} a^{6} + \frac{17}{4} a^{5} - \frac{3}{4} a^{4} + \frac{11}{4} a^{3} - \frac{5}{4} a^{2} + \frac{25}{8} a - \frac{7}{8} \),  \( \frac{3}{4} a^{6} + \frac{7}{2} a^{4} + \frac{5}{2} a^{2} + \frac{13}{4} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 4.5001159868 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_4$ (as 8T4):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 8
The 5 conjugacy class representatives for $D_4$
Character table for $D_4$

Intermediate fields

\(\Q(\sqrt{-17}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{17}) \), \(\Q(i, \sqrt{17})\), 4.0.272.1 x2, 4.2.1156.1 x2

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

Sibling fields

Degree 4 siblings: 4.2.1156.1, 4.0.272.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.17.2t1.1c1$1$ $ 17 $ $x^{2} - x - 4$ $C_2$ (as 2T1) $1$ $1$
1.2e2.2t1.1c1$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
1.2e2_17.2t1.1c1$1$ $ 2^{2} \cdot 17 $ $x^{2} + 17$ $C_2$ (as 2T1) $1$ $-1$
2.2e2_17.4t3.3c1$2$ $ 2^{2} \cdot 17 $ $x^{8} + 5 x^{6} + 4 x^{4} + 5 x^{2} + 1$ $D_4$ (as 8T4) $1$ $0$

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 *.