# Properties

 Label 8.0.2070185663499849.1 Degree $8$ Signature $[0, 4]$ Discriminant $3^{6}\cdot 7^{6}\cdot 17^{6}$ Root discriminant $82.13$ Ramified primes $3, 7, 17$ Class number $72$ Class group $[2, 6, 6]$ Galois group $Q_8$ (as 8T5)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 65*x^6 - 439*x^5 + 1876*x^4 - 12191*x^3 + 60887*x^2 - 124718*x + 121291)

gp: K = bnfinit(x^8 - x^7 + 65*x^6 - 439*x^5 + 1876*x^4 - 12191*x^3 + 60887*x^2 - 124718*x + 121291, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121291, -124718, 60887, -12191, 1876, -439, 65, -1, 1]);

## Normalizeddefining polynomial

$$x^{8} - x^{7} + 65 x^{6} - 439 x^{5} + 1876 x^{4} - 12191 x^{3} + 60887 x^{2} - 124718 x + 121291$$

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: $$2070185663499849=3^{6}\cdot 7^{6}\cdot 17^{6}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $82.13$ 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: $3, 7, 17$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Gal(K/\Q)|$: $8$ This field is Galois over $\Q$. This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{502} a^{6} + \frac{20}{251} a^{5} - \frac{83}{251} a^{4} + \frac{243}{502} a^{3} + \frac{141}{502} a^{2} - \frac{227}{502} a + \frac{115}{502}$, $\frac{1}{467748573634} a^{7} + \frac{165358440}{233874286817} a^{6} + \frac{44501174544}{233874286817} a^{5} - \frac{198425690701}{467748573634} a^{4} + \frac{58248275037}{467748573634} a^{3} - \frac{147684398015}{467748573634} a^{2} + \frac{227073005275}{467748573634} a + \frac{41783659487}{233874286817}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

$C_{2}\times C_{6}\times C_{6}$, which has order $72$

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: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$154.472027483$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$Q_8$ (as 8T5):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 8 The 5 conjugacy class representatives for $Q_8$ Character table for $Q_8$

## Intermediate fields

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

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2} 77.8.6.1x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$17$17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4} 17.4.3.1x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.21.2t1.a.a$1$ $3 \cdot 7$ $x^{2} - x - 5$ $C_2$ (as 2T1) $1$ $1$
* 1.357.2t1.a.a$1$ $3 \cdot 7 \cdot 17$ $x^{2} - x - 89$ $C_2$ (as 2T1) $1$ $1$
* 1.17.2t1.a.a$1$ $17$ $x^{2} - x - 4$ $C_2$ (as 2T1) $1$ $1$
*2 2.127449.8t5.b.a$2$ $3^{2} \cdot 7^{2} \cdot 17^{2}$ $x^{8} - x^{7} + 65 x^{6} - 439 x^{5} + 1876 x^{4} - 12191 x^{3} + 60887 x^{2} - 124718 x + 121291$ $Q_8$ (as 8T5) $-1$ $-2$

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 *.