Properties

 Label 8.0.65804544.1 Degree $8$ Signature $[0, 4]$ Discriminant $2^{8}\cdot 3^{2}\cdot 13^{4}$ Root discriminant $9.49$ Ramified primes $2, 3, 13$ Class number $1$ Class group Trivial Galois group $C_2^2 \wr C_2$ (as 8T18)

Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^8 + x^6 - x^4 + x^2 + 1)

gp: K = bnfinit(x^8 + x^6 - x^4 + x^2 + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1, 0, -1, 0, 1, 0, 1]);

Normalizeddefining polynomial

$$x^{8} + x^{6} - x^{4} + x^{2} + 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

Invariants

 Degree: $8$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 4]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$65804544=2^{8}\cdot 3^{2}\cdot 13^{4}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $9.49$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $2, 3, 13$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $4$ This field is not Galois over $\Q$. This is not a CM field.

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

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

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $3$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$a^{7} + a^{5} - a^{3} + a$$,  $$a^{7} - a^{3} + 2 a$$,  $$a^{7} + a^{6} + 2 a^{5} + a^{4} - a^{3} - 2 a^{2} - 2 a - 2$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$5.55524804914$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

Galois group

$C_2^2\wr C_2$ (as 8T18):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 32 The 14 conjugacy class representatives for $C_2^2 \wr C_2$ Character table for $C_2^2 \wr C_2$

Intermediate fields

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 siblings: data not computed Degree 16 siblings: 16.0.89791815397090000896.2, 16.0.350749278894882816.3, 16.8.89791815397090000896.1, 16.0.89791815397090000896.9, 16.0.531312517142544384.3, 16.0.1108540930828271616.1, 16.0.89791815397090000896.3

Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])

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];

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.8.2$x^{8} + 2 x^{7} + 8 x^{2} + 48$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{4} 33.2.1.1x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2} 3.2.0.1x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2} 1313.4.2.1x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.12.2t1.a.a$1$ $2^{2} \cdot 3$ $x^{2} - 3$ $C_2$ (as 2T1) $1$ $1$
1.39.2t1.a.a$1$ $3 \cdot 13$ $x^{2} - x + 10$ $C_2$ (as 2T1) $1$ $-1$
1.52.2t1.a.a$1$ $2^{2} \cdot 13$ $x^{2} + 13$ $C_2$ (as 2T1) $1$ $-1$
1.156.2t1.a.a$1$ $2^{2} \cdot 3 \cdot 13$ $x^{2} - 39$ $C_2$ (as 2T1) $1$ $1$
* 1.13.2t1.a.a$1$ $13$ $x^{2} - x - 3$ $C_2$ (as 2T1) $1$ $1$
1.4.2t1.a.a$1$ $2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
1.3.2t1.a.a$1$ $3$ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
2.1872.4t3.a.a$2$ $2^{4} \cdot 3^{2} \cdot 13$ $x^{4} - 9 x^{2} - 9$ $D_{4}$ (as 4T3) $1$ $0$
* 2.624.4t3.g.a$2$ $2^{4} \cdot 3 \cdot 13$ $x^{4} + 5 x^{2} + 3$ $D_{4}$ (as 4T3) $1$ $-2$
* 2.39.4t3.a.a$2$ $3 \cdot 13$ $x^{4} - x^{3} - x^{2} + x + 1$ $D_{4}$ (as 4T3) $1$ $0$
* 2.208.4t3.b.a$2$ $2^{4} \cdot 13$ $x^{4} - 3 x^{2} - 1$ $D_{4}$ (as 4T3) $1$ $0$
2.624.4t3.b.a$2$ $2^{4} \cdot 3 \cdot 13$ $x^{4} + 7 x^{2} + 13$ $D_{4}$ (as 4T3) $1$ $0$
2.624.4t3.f.a$2$ $2^{4} \cdot 3 \cdot 13$ $x^{4} - 5 x^{2} + 3$ $D_{4}$ (as 4T3) $1$ $2$

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 *.