# Properties

 Label 8.2.14301947824.1 Degree $8$ Signature $[2, 3]$ Discriminant $-14301947824$ Root discriminant $18.60$ Ramified primes $2, 19$ Class number $1$ Class group trivial Galois group $\textrm{GL(2,3)}$ (as 8T23)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

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

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, -6, 4, 3, 2, -7, 7, -4, 1]);

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

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: $$-14301947824$$$$\medspace = -\,2^{4}\cdot 19^{7}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $18.60$ 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: $2, 19$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} + \frac{2}{7} a^{3} - \frac{3}{7} a^{2} - \frac{1}{7} a - \frac{3}{7}$

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: $4$ 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: $$a^{3} - 3 a^{2} + 2 a + 1$$,  $$a^{3} - a - 1$$,  $$\frac{1}{7} a^{7} - \frac{5}{7} a^{3} - \frac{10}{7} a^{2} - \frac{8}{7} a - \frac{3}{7}$$,  $$\frac{1}{7} a^{7} - a^{5} + 3 a^{4} - \frac{33}{7} a^{3} + \frac{25}{7} a^{2} - \frac{1}{7} a - \frac{3}{7}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$315.475807248$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{3}\cdot 315.475807248 \cdot 1}{2\sqrt{14301947824}}\approx 1.30869386467$

## 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: 8.2.14301947824.2

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R ${\href{/padicField/3.8.0.1}{8} }$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ R ${\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }$ ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }$ ${\href{/padicField/41.8.0.1}{8} }$ ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2} 2.3.2.1x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2} 1919.8.7.2x^{8} - 19$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
1.19.2t1.a.a$1$ $19$ $$\Q(\sqrt{-19})$$ $C_2$ (as 2T1) $1$ $-1$
2.76.3t2.a.a$2$ $2^{2} \cdot 19$ 3.1.76.1 $S_3$ (as 3T2) $1$ $0$
2.1444.24t22.a.a$2$ $2^{2} \cdot 19^{2}$ 8.2.14301947824.1 $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
2.1444.24t22.a.b$2$ $2^{2} \cdot 19^{2}$ 8.2.14301947824.1 $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
3.1444.6t8.a.a$3$ $2^{2} \cdot 19^{2}$ 4.2.27436.1 $S_4$ (as 4T5) $1$ $-1$
* 3.27436.4t5.a.a$3$ $2^{2} \cdot 19^{3}$ 4.2.27436.1 $S_4$ (as 4T5) $1$ $1$
* 4.521284.8t23.a.a$4$ $2^{2} \cdot 19^{4}$ 8.2.14301947824.1 $\textrm{GL(2,3)}$ (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 *.