# Properties

 Label 8.0.2178981.1 Degree $8$ Signature $[0, 4]$ Discriminant $3^{6}\cdot 7^{2}\cdot 61$ Root discriminant $6.20$ Ramified primes $3, 7, 61$ Class number $1$ Class group Trivial Galois Group $C_2 \wr C_2\wr C_2$ (as 8T35)

# Learn more about

Show commands for: Magma / SageMath / Pari/GP

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

## Normalizeddefining polynomial

$$x^{8}$$ $$\mathstrut -\mathstrut x^{7}$$ $$\mathstrut -\mathstrut x^{6}$$ $$\mathstrut +\mathstrut 2 x^{5}$$ $$\mathstrut +\mathstrut x^{4}$$ $$\mathstrut -\mathstrut 2 x^{3}$$ $$\mathstrut -\mathstrut x^{2}$$ $$\mathstrut +\mathstrut x$$ $$\mathstrut +\mathstrut 1$$

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: $$2178981=3^{6}\cdot 7^{2}\cdot 61$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $6.20$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $3, 7, 61$ 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 $$a$$)

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

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: $$a^{5} - a^{4} + a^{2}$$ (order $6$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental units: $$a$$,  $$a^{6} - a^{5} - a^{4} + 2 a^{3} - a - 1$$,  $$a^{7} - 2 a^{6} + 3 a^{4} - a^{3} - 2 a^{2} + 2$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$1.34805397566$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

$D_4^2.C_2$ (as 8T35):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A solvable group of order 128 The 20 conjugacy class representatives for $C_2 \wr C_2\wr C_2$ Character table for $C_2 \wr C_2\wr C_2$

## Intermediate fields

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

## Sibling fields

 Degree 8 siblings: data not computed Degree 16 siblings: data not computed Degree 32 siblings: data not computed

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.8.0.1}{8} }$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2} 77.4.2.1x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4} 61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2} 61.2.0.1x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.7.2t1.1c1$1$ $7$ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
* 1.3.2t1.1c1$1$ $3$ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.3_7.2t1.1c1$1$ $3 \cdot 7$ $x^{2} - x - 5$ $C_2$ (as 2T1) $1$ $1$
1.7_61.2t1.1c1$1$ $7 \cdot 61$ $x^{2} - x + 107$ $C_2$ (as 2T1) $1$ $-1$
1.61.2t1.1c1$1$ $61$ $x^{2} - x - 15$ $C_2$ (as 2T1) $1$ $1$
1.3_7_61.2t1.1c1$1$ $3 \cdot 7 \cdot 61$ $x^{2} - x - 320$ $C_2$ (as 2T1) $1$ $1$
1.3_61.2t1.1c1$1$ $3 \cdot 61$ $x^{2} - x + 46$ $C_2$ (as 2T1) $1$ $-1$
2.3e2_7_61.4t3.3c1$2$ $3^{2} \cdot 7 \cdot 61$ $x^{4} - 2 x^{3} - 15 x^{2} + 16 x + 76$ $D_{4}$ (as 4T3) $1$ $0$
2.3e2_7_61.4t3.4c1$2$ $3^{2} \cdot 7 \cdot 61$ $x^{4} - x^{3} + 18 x^{2} - 13 x + 79$ $D_{4}$ (as 4T3) $1$ $0$
2.3_7e2_61.4t3.2c1$2$ $3 \cdot 7^{2} \cdot 61$ $x^{4} - 2 x^{3} + 26 x^{2} - 25 x + 193$ $D_{4}$ (as 4T3) $1$ $0$
2.3_61.4t3.2c1$2$ $3 \cdot 61$ $x^{4} - 2 x^{3} - 2 x^{2} + 3 x + 3$ $D_{4}$ (as 4T3) $1$ $0$
* 2.3e2_7.4t3.2c1$2$ $3^{2} \cdot 7$ $x^{4} - x^{3} + 2 x + 1$ $D_{4}$ (as 4T3) $1$ $0$
2.3e2_7_61e2.4t3.2c1$2$ $3^{2} \cdot 7 \cdot 61^{2}$ $x^{4} - x^{3} + 138 x^{2} - 46 x + 4861$ $D_{4}$ (as 4T3) $1$ $0$
4.3e3_7e2_61e2.8t29.3c1$4$ $3^{3} \cdot 7^{2} \cdot 61^{2}$ $x^{8} - x^{7} - 9 x^{6} + 8 x^{5} + 25 x^{4} - 24 x^{3} - 18 x^{2} + 30 x - 3$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $1$ $2$
4.3e3_7e3_61.8t35.2c1$4$ $3^{3} \cdot 7^{3} \cdot 61$ $x^{8} - x^{7} - x^{6} + 2 x^{5} + x^{4} - 2 x^{3} - x^{2} + x + 1$ $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $0$
4.3e3_7_61e3.8t35.2c1$4$ $3^{3} \cdot 7 \cdot 61^{3}$ $x^{8} - x^{7} - x^{6} + 2 x^{5} + x^{4} - 2 x^{3} - x^{2} + x + 1$ $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $0$
4.3e3_7e2_61e2.8t31.1c1$4$ $3^{3} \cdot 7^{2} \cdot 61^{2}$ $x^{8} - x^{7} - 9 x^{6} + 8 x^{5} + 25 x^{4} - 24 x^{3} - 18 x^{2} + 30 x - 3$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $1$ $-2$
4.3e3_7e3_61e3.8t35.2c1$4$ $3^{3} \cdot 7^{3} \cdot 61^{3}$ $x^{8} - x^{7} - x^{6} + 2 x^{5} + x^{4} - 2 x^{3} - x^{2} + x + 1$ $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $0$
* 4.3e3_7_61.8t35.2c1$4$ $3^{3} \cdot 7 \cdot 61$ $x^{8} - x^{7} - x^{6} + 2 x^{5} + x^{4} - 2 x^{3} - x^{2} + x + 1$ $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $0$

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 *.