# Properties

 Label 8.0.190563597.4 Degree $8$ Signature $[0, 4]$ Discriminant $3^{4}\cdot 7^{3}\cdot 19^{3}$ Root discriminant $10.84$ Ramified primes $3, 7, 19$ Class number $1$ Class group Trivial Galois Group $D_{8}$ (as 8T6)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -9, -3, 6, 1, -6, 7, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 + 7*x^6 - 6*x^5 + x^4 + 6*x^3 - 3*x^2 - 9*x + 9)
gp: K = bnfinit(x^8 - 3*x^7 + 7*x^6 - 6*x^5 + x^4 + 6*x^3 - 3*x^2 - 9*x + 9, 1)

## Normalizeddefining polynomial

$$x^{8}$$ $$\mathstrut -\mathstrut 3 x^{7}$$ $$\mathstrut +\mathstrut 7 x^{6}$$ $$\mathstrut -\mathstrut 6 x^{5}$$ $$\mathstrut +\mathstrut x^{4}$$ $$\mathstrut +\mathstrut 6 x^{3}$$ $$\mathstrut -\mathstrut 3 x^{2}$$ $$\mathstrut -\mathstrut 9 x$$ $$\mathstrut +\mathstrut 9$$

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: $$190563597=3^{4}\cdot 7^{3}\cdot 19^{3}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $10.84$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $3, 7, 19$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$

magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk

## Class group and class number

Trivial Abelian group, 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: $$\frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{2}{3} a^{3} + \frac{1}{3} a^{2}$$ (order $6$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental units: $$\frac{1}{9} a^{7} - \frac{1}{3} a^{6} + \frac{1}{9} a^{5} - \frac{1}{3} a^{4} - \frac{8}{9} a^{3} - \frac{2}{3} a^{2} + \frac{1}{3} a$$,  $$\frac{1}{9} a^{7} - \frac{2}{9} a^{5} + \frac{4}{3} a^{4} - \frac{14}{9} a^{3} + \frac{4}{3} a^{2} + \frac{1}{3} a$$,  $$\frac{2}{9} a^{7} - \frac{2}{3} a^{6} + \frac{8}{9} a^{5} - \frac{1}{3} a^{4} - \frac{13}{9} a^{3} + \frac{4}{3} a^{2} + \frac{2}{3} a - 1$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$50.7926829502$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

$D_8$ (as 8T6):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A solvable group of order 16 The 7 conjugacy class representatives for $D_{8}$ Character table for $D_{8}$

## 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 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 ${\href{/LocalNumberField/2.8.0.1}{8} }$ R ${\href{/LocalNumberField/5.8.0.1}{8} }$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2} 3.2.1.2x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2} 3.2.1.2x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2} 7.2.1.1x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$V_4$$[\ ]_{2}^{2} 1919.2.1.2x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2} 19.4.2.1x^{4} + 57 x^{2} + 1444$$2$$2$$2$$V_4$$[\ ]_{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_19.2t1.1c1$1$ $7 \cdot 19$ $x^{2} - x - 33$ $C_2$ (as 2T1) $1$ $1$
* 1.3.2t1.1c1$1$ $3$ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.3_7_19.2t1.1c1$1$ $3 \cdot 7 \cdot 19$ $x^{2} - x + 100$ $C_2$ (as 2T1) $1$ $-1$
* 2.3_7_19.4t3.4c1$2$ $3 \cdot 7 \cdot 19$ $x^{4} - x^{3} - 5 x^{2} + 3 x + 9$ $D_{4}$ (as 4T3) $1$ $0$
* 2.3_7_19.8t6.3c1$2$ $3 \cdot 7 \cdot 19$ $x^{8} - 3 x^{7} + 7 x^{6} - 6 x^{5} + x^{4} + 6 x^{3} - 3 x^{2} - 9 x + 9$ $D_{8}$ (as 8T6) $1$ $0$
* 2.3_7_19.8t6.3c2$2$ $3 \cdot 7 \cdot 19$ $x^{8} - 3 x^{7} + 7 x^{6} - 6 x^{5} + x^{4} + 6 x^{3} - 3 x^{2} - 9 x + 9$ $D_{8}$ (as 8T6) $1$ $0$

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 *.