# Properties

 Label 8.2.265847707.1 Degree $8$ Signature $[2, 3]$ Discriminant $-265847707$ Root discriminant $11.30$ Ramified prime $643$ Class number $1$ Class group trivial Galois group $\textrm{GL(2,3)}$ (as 8T23)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + 3*x^6 + x^5 - 7*x^4 + 18*x^3 - 12*x^2 + 4*x - 1)

gp: K = bnfinit(x^8 - 2*x^7 + 3*x^6 + x^5 - 7*x^4 + 18*x^3 - 12*x^2 + 4*x - 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 4, -12, 18, -7, 1, 3, -2, 1]);

$$x^{8} - 2 x^{7} + 3 x^{6} + x^{5} - 7 x^{4} + 18 x^{3} - 12 x^{2} + 4 x - 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $8$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[2, 3]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-265847707$$$$\medspace = -\,643^{3}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $11.30$ 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: $643$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{127} a^{7} - \frac{14}{127} a^{6} + \frac{44}{127} a^{5} - \frac{19}{127} a^{4} - \frac{33}{127} a^{3} + \frac{33}{127} a^{2} - \frac{27}{127} a - \frac{53}{127}$

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: $4$ 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: $$10.712578457$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{3}\cdot 10.712578457 \cdot 1}{2\sqrt{265847707}}\approx 0.32594712670$

## Galois group

$\GL(2,3)$ (as 8T23):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 48 The 8 conjugacy class representatives for $\textrm{GL(2,3)}$ Character table for $\textrm{GL(2,3)}$

## Intermediate fields

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

## Sibling fields

 Degree 16 sibling: Deg 16 Degree 24 sibling: Deg 24 Arithmetically equvalently sibling: 8.2.265847707.2

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }$ ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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
643Data not computed

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.643.2t1.a.a$1$ $643$ $x^{2} - x + 161$ $C_2$ (as 2T1) $1$ $-1$
2.643.3t2.a.a$2$ $643$ $x^{3} - 2 x - 5$ $S_3$ (as 3T2) $1$ $0$
2.643.24t22.a.a$2$ $643$ $x^{8} - 2 x^{7} + 3 x^{6} + x^{5} - 7 x^{4} + 18 x^{3} - 12 x^{2} + 4 x - 1$ $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
2.643.24t22.a.b$2$ $643$ $x^{8} - 2 x^{7} + 3 x^{6} + x^{5} - 7 x^{4} + 18 x^{3} - 12 x^{2} + 4 x - 1$ $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
3.413449.6t8.a.a$3$ $643^{2}$ $x^{4} - x^{3} - 2 x + 1$ $S_4$ (as 4T5) $1$ $-1$
* 3.643.4t5.a.a$3$ $643$ $x^{4} - x^{3} - 2 x + 1$ $S_4$ (as 4T5) $1$ $1$
* 4.413449.8t23.a.a$4$ $643^{2}$ $x^{8} - 2 x^{7} + 3 x^{6} + x^{5} - 7 x^{4} + 18 x^{3} - 12 x^{2} + 4 x - 1$ $\textrm{GL(2,3)}$ (as 8T23) $1$ $0$

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 *.