# Properties

 Label 8.0.1797309.1 Degree $8$ Signature $[0, 4]$ Discriminant $3^{4}\cdot 22189$ Root discriminant $6.05$ Ramified primes $3, 22189$ Class number $1$ Class group Trivial Galois group $S_4\wr C_2$ (as 8T47)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 7, -9, 10, -8, 6, -3, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 + 6*x^6 - 8*x^5 + 10*x^4 - 9*x^3 + 7*x^2 - 4*x + 1)

gp: K = bnfinit(x^8 - 3*x^7 + 6*x^6 - 8*x^5 + 10*x^4 - 9*x^3 + 7*x^2 - 4*x + 1, 1)

## Normalizeddefining polynomial

$$x^{8} - 3 x^{7} + 6 x^{6} - 8 x^{5} + 10 x^{4} - 9 x^{3} + 7 x^{2} - 4 x + 1$$

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: $$1797309=3^{4}\cdot 22189$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $6.05$ magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));  sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $3, 22189$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ $|\Aut(K/\Q)|$: $1$ 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: $$2 a^{7} - 5 a^{6} + 10 a^{5} - 12 a^{4} + 15 a^{3} - 11 a^{2} + 9 a - 3$$ (order $6$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);  sage: UK.torsion_generator()  gp: K.tu[2] Fundamental units: $$a^{7} - 2 a^{6} + 4 a^{5} - 5 a^{4} + 7 a^{3} - 5 a^{2} + 4 a - 2$$,  $$2 a^{7} - 5 a^{6} + 9 a^{5} - 10 a^{4} + 13 a^{3} - 10 a^{2} + 8 a - 3$$,  $$2 a^{7} - 4 a^{6} + 7 a^{5} - 7 a^{4} + 10 a^{3} - 6 a^{2} + 6 a - 2$$ magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$1.20014144807$$ magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

$S_4\wr C_2$ (as 8T47):

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A solvable group of order 1152 The 20 conjugacy class representatives for $S_4\wr C_2$ Character table for $S_4\wr C_2$

## Intermediate fields

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

## Sibling fields

 Degree 12 siblings: data not computed Degree 16 siblings: data not computed Degree 18 siblings: data not computed Degree 24 siblings: data not computed Degree 32 siblings: data not computed Degree 36 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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
22189Data not computed