# Properties

 Label 8.0.3240000.1 Degree $8$ Signature $[0, 4]$ Discriminant $2^{6}\cdot 3^{4}\cdot 5^{4}$ Root discriminant $6.51$ Ramified primes $2, 3, 5$ Class number $1$ Class group Trivial Galois Group $Q_8:C_2$ (as 8T11)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

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

## Normalizeddefining polynomial

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

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: $$3240000=2^{6}\cdot 3^{4}\cdot 5^{4}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $6.51$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 3, 5$ 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: $$-a^{6} + 2 a^{5} - 2 a^{3} - a^{2} + 4 a - 1$$ (order $6$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental units: $$a^{7} - 3 a^{6} + 2 a^{5} + 2 a^{4} - a^{3} - 5 a^{2} + 6 a - 2$$,  $$a^{7} - 2 a^{6} + a^{5} + a^{4} - 2 a^{2} + 3 a - 2$$,  $$a^{7} - 2 a^{6} + 2 a^{4} + a^{3} - 4 a^{2} + a$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$1.89812936734$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

$D_4:C_2$ (as 8T11):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A solvable group of order 16 The 10 conjugacy class representatives for $Q_8:C_2$ Character table for $Q_8:C_2$

## 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

## 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 R ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2} 2.4.0.1x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4} 55.8.4.1x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5.2t1.1c1$1$ $5$ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.2e3_3.2t1.2c1$1$ $2^{3} \cdot 3$ $x^{2} + 6$ $C_2$ (as 2T1) $1$ $-1$
1.2e3_3_5.2t1.2c1$1$ $2^{3} \cdot 3 \cdot 5$ $x^{2} + 30$ $C_2$ (as 2T1) $1$ $-1$
1.2e3_5.2t1.1c1$1$ $2^{3} \cdot 5$ $x^{2} - 10$ $C_2$ (as 2T1) $1$ $1$
* 1.3.2t1.1c1$1$ $3$ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.2e3.2t1.1c1$1$ $2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
* 1.3_5.2t1.1c1$1$ $3 \cdot 5$ $x^{2} - x + 4$ $C_2$ (as 2T1) $1$ $-1$
* 2.2e3_3_5.8t11.2c1$2$ $2^{3} \cdot 3 \cdot 5$ $x^{8} - 3 x^{7} + 3 x^{6} + x^{5} - 2 x^{4} - 3 x^{3} + 7 x^{2} - 4 x + 1$ $Q_8:C_2$ (as 8T11) $0$ $0$
* 2.2e3_3_5.8t11.2c2$2$ $2^{3} \cdot 3 \cdot 5$ $x^{8} - 3 x^{7} + 3 x^{6} + x^{5} - 2 x^{4} - 3 x^{3} + 7 x^{2} - 4 x + 1$ $Q_8:C_2$ (as 8T11) $0$ $0$

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 *.