# Properties

 Label 8.0.1385533.1 Degree $8$ Signature $[0, 4]$ Discriminant $29\cdot 47777$ Root discriminant $5.86$ Ramified primes $29, 47777$ Class number $1$ Class group Trivial Galois Group $S_8$ (as 8T50)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 1, -3, 3, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + 3*x^6 - 3*x^5 + x^4 + 1)
gp: K = bnfinit(x^8 - 2*x^7 + 3*x^6 - 3*x^5 + x^4 + 1, 1)

## Normalizeddefining polynomial

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

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: $$1385533=29\cdot 47777$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $5.86$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $29, 47777$ 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: $$-1$$ (order $2$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental units: $$a^{7} - 2 a^{6} + 2 a^{5} - 2 a^{4} + a^{3} + a$$,  $$a^{7} - 2 a^{6} + 3 a^{5} - 3 a^{4} + a^{3}$$,  $$a^{7} - 2 a^{6} + 2 a^{5} - 2 a^{4} + a^{2}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$0.331112529152$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

$S_8$ (as 8T50):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A non-solvable group of order 40320 The 22 conjugacy class representatives for $S_8$ Character table for $S_8$ is not computed

## Intermediate fields

 The extension is primitive: there are no intermediate fields between this field and $\Q$.

## Sibling fields

 Degree 16 sibling: data not computed Degree 28 sibling: data not computed Degree 30 sibling: data not computed Degree 35 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }$ ${\href{/LocalNumberField/3.5.0.1}{5} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ R ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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

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
$29$$\Q_{29}$$x + 2$$1$$1$$0Trivial[\ ] \Q_{29}$$x + 2$$1$$1$$0Trivial[\ ] 29.2.1.1x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
47777Data not computed