# Properties

 Label 8.0.4102893.1 Degree $8$ Signature $[0, 4]$ Discriminant $3^{4}\cdot 37^{3}$ Root discriminant $6.71$ Ramified primes $3, 37$ Class number $1$ Class group Trivial Galois group $D_{8}$ (as 8T6)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

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

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 6, -6, 5, -3, 3, -3, 1]);

## Normalizeddefining polynomial

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

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $8$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 4]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$4102893=3^{4}\cdot 37^{3}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $6.71$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $3, 37$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $2$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

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

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $3$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-2 a^{7} + 5 a^{6} - 3 a^{5} + 3 a^{4} - 8 a^{3} + 8 a^{2} - 6 a + 2$$ (order $6$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$2 a^{7} - 4 a^{6} + a^{5} - 3 a^{4} + 7 a^{3} - 4 a^{2} + 4 a$$,  $$a - 1$$,  $$a^{7} - a^{6} - 2 a^{5} + 2 a^{3} + 2 a^{2} - 2 a + 2$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$2.02888107198$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$D_8$ (as 8T6):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 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: 16.0.23045377697175681.1 Degree 8 sibling: 8.2.50602347.1

## 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 ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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

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]])

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];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} 3.4.2.1x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$37$37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2} 37.2.1.2x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3.2t1.a.a$1$ $3$ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.37.2t1.a.a$1$ $37$ $x^{2} - x - 9$ $C_2$ (as 2T1) $1$ $1$
1.111.2t1.a.a$1$ $3 \cdot 37$ $x^{2} - x + 28$ $C_2$ (as 2T1) $1$ $-1$
* 2.111.4t3.a.a$2$ $3 \cdot 37$ $x^{4} - x^{3} - 2 x^{2} + 3$ $D_{4}$ (as 4T3) $1$ $0$
* 2.111.8t6.a.a$2$ $3 \cdot 37$ $x^{8} - 3 x^{7} + 3 x^{6} - 3 x^{5} + 5 x^{4} - 6 x^{3} + 6 x^{2} - 3 x + 1$ $D_{8}$ (as 8T6) $1$ $0$
* 2.111.8t6.a.b$2$ $3 \cdot 37$ $x^{8} - 3 x^{7} + 3 x^{6} - 3 x^{5} + 5 x^{4} - 6 x^{3} + 6 x^{2} - 3 x + 1$ $D_{8}$ (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 *.