# Properties

 Label 8.2.8671400783.1 Degree $8$ Signature $[2, 3]$ Discriminant $-8671400783$ Root discriminant $$17.47$$ Ramified primes see page Class number $1$ Class group trivial Galois group $D_{8}$ (as 8T6)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 2*x^6 + x^5 - 15*x^4 + 23*x^3 + 6*x^2 - 27*x + 19)

gp: K = bnfinit(x^8 - x^7 + 2*x^6 + x^5 - 15*x^4 + 23*x^3 + 6*x^2 - 27*x + 19, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, -27, 6, 23, -15, 1, 2, -1, 1]);

$$x^{8} - x^{7} + 2x^{6} + x^{5} - 15x^{4} + 23x^{3} + 6x^{2} - 27x + 19$$

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: $$-8671400783$$ -8671400783 $$\medspace = -\,17^{4}\cdot 47^{3}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $$17.47$$ 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: $$17$$, $$47$$ 17, 47 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $\card{ \Aut(K/\Q) }$: $2$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{273}a^{7}-\frac{4}{273}a^{6}+\frac{2}{39}a^{5}-\frac{44}{91}a^{4}+\frac{17}{273}a^{3}-\frac{17}{39}a^{2}-\frac{92}{273}a-\frac{8}{91}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

 Monogenic: No Index: Not computed Inessential primes: $3$

## 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: $$-1$$ -1  (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $\frac{12}{91}a^{7}-\frac{53}{273}a^{6}+\frac{7}{39}a^{5}-\frac{20}{273}a^{4}-\frac{571}{273}a^{3}+\frac{43}{13}a^{2}+\frac{328}{273}a-\frac{379}{91}$, $\frac{2}{273}a^{7}-\frac{8}{273}a^{6}+\frac{4}{39}a^{5}+\frac{3}{91}a^{4}+\frac{34}{273}a^{3}+\frac{5}{39}a^{2}-\frac{184}{273}a+\frac{75}{91}$, $\frac{16}{273}a^{7}+\frac{9}{91}a^{6}+\frac{2}{13}a^{5}+\frac{163}{273}a^{4}-\frac{92}{273}a^{3}+\frac{1}{39}a^{2}+\frac{530}{273}a-\frac{128}{91}$, $\frac{22}{273}a^{7}+\frac{1}{91}a^{6}+\frac{5}{39}a^{5}+\frac{33}{91}a^{4}-\frac{118}{91}a^{3}+\frac{29}{39}a^{2}+\frac{524}{273}a-\frac{346}{273}$ 12/91*a^7 - 53/273*a^6 + 7/39*a^5 - 20/273*a^4 - 571/273*a^3 + 43/13*a^2 + 328/273*a - 379/91, 2/273*a^7 - 8/273*a^6 + 4/39*a^5 + 3/91*a^4 + 34/273*a^3 + 5/39*a^2 - 184/273*a + 75/91, 16/273*a^7 + 9/91*a^6 + 2/13*a^5 + 163/273*a^4 - 92/273*a^3 + 1/39*a^2 + 530/273*a - 128/91, 22/273*a^7 + 1/91*a^6 + 5/39*a^5 + 33/91*a^4 - 118/91*a^3 + 29/39*a^2 + 524/273*a - 346/273 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$81.706186763$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{3}\cdot 81.706186763 \cdot 1}{2\sqrt{8671400783}}\approx 0.43529122614$

## 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: 16.0.166101760110563345913601.2 Degree 8 sibling: 8.0.23973872753.1

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/padicField/2.4.0.1}{4} }^{2}$ ${\href{/padicField/3.2.0.1}{2} }^{4}$ ${\href{/padicField/5.8.0.1}{8} }$ ${\href{/padicField/7.2.0.1}{2} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }$ ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ R ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }$ ${\href{/padicField/29.8.0.1}{8} }$ ${\href{/padicField/31.8.0.1}{8} }$ ${\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }$ ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ R ${\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{2}$

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
$$17$$ 17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2} 17.2.1.1x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2} 17.2.1.1x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
$$47$$ $\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2} 47.2.1.2x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
* 1.17.2t1.a.a$1$ $17$ $$\Q(\sqrt{17})$$ $C_2$ (as 2T1) $1$ $1$
1.47.2t1.a.a$1$ $47$ $$\Q(\sqrt{-47})$$ $C_2$ (as 2T1) $1$ $-1$
1.799.2t1.a.a$1$ $17 \cdot 47$ $$\Q(\sqrt{-799})$$ $C_2$ (as 2T1) $1$ $-1$
* 2.799.4t3.a.a$2$ $17 \cdot 47$ 4.2.13583.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.799.8t6.a.a$2$ $17 \cdot 47$ 8.2.8671400783.1 $D_{8}$ (as 8T6) $1$ $0$
* 2.799.8t6.a.b$2$ $17 \cdot 47$ 8.2.8671400783.1 $D_{8}$ (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 *.