# Properties

 Label 8.8.2070185663499849.1 Degree $8$ Signature $[8, 0]$ Discriminant $2.070\times 10^{15}$ Root discriminant $82.13$ Ramified primes $3, 7, 17$ Class number $2$ Class group $[2]$ Galois group $Q_8$ (as 8T5)

# Learn more

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 - 110*x^6 + 153*x^5 + 3789*x^4 + 1989*x^3 - 44000*x^2 - 97899*x - 46703)

gp: K = bnfinit(x^8 - 3*x^7 - 110*x^6 + 153*x^5 + 3789*x^4 + 1989*x^3 - 44000*x^2 - 97899*x - 46703, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-46703, -97899, -44000, 1989, 3789, 153, -110, -3, 1]);

$$x^{8} - 3 x^{7} - 110 x^{6} + 153 x^{5} + 3789 x^{4} + 1989 x^{3} - 44000 x^{2} - 97899 x - 46703$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $8$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[8, 0]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$2070185663499849$$$$\medspace = 3^{6}\cdot 7^{6}\cdot 17^{6}$$ 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))); $\card{ \Gal(K/\Q) }$: $8$ This field is Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{1427439851} a^{7} - \frac{530654460}{1427439851} a^{6} + \frac{229561142}{1427439851} a^{5} - \frac{498841993}{1427439851} a^{4} + \frac{376200939}{1427439851} a^{3} - \frac{286450262}{1427439851} a^{2} - \frac{191560992}{1427439851} a + \frac{415464251}{1427439851}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

$C_{2}$, which has order $2$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $7$ 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: $$36250.7464952$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{0}\cdot 36250.7464952 \cdot 2}{2\sqrt{2070185663499849}}\approx 0.203963413295$

## 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

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/padicField/2.4.0.1}{4} }^{2}$ R ${\href{/padicField/5.4.0.1}{4} }^{2}$ R ${\href{/padicField/11.4.0.1}{4} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}$ R ${\href{/padicField/19.4.0.1}{4} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{4}$ ${\href{/padicField/47.1.0.1}{1} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{4}$

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

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$17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4} 17.4.3.2x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
* 1.21.2t1.a.a$1$ $3 \cdot 7$ $$\Q(\sqrt{21})$$ $C_2$ (as 2T1) $1$ $1$
* 1.17.2t1.a.a$1$ $17$ $$\Q(\sqrt{17})$$ $C_2$ (as 2T1) $1$ $1$
* 1.357.2t1.a.a$1$ $3 \cdot 7 \cdot 17$ $$\Q(\sqrt{357})$$ $C_2$ (as 2T1) $1$ $1$
*2 2.127449.8t5.a.a$2$ $3^{2} \cdot 7^{2} \cdot 17^{2}$ 8.8.2070185663499849.1 $Q_8$ (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 *.