# Properties

 Label 8.2.107171875.1 Degree $8$ Signature $[2, 3]$ Discriminant $-107171875$ Root discriminant $$10.09$$ Ramified primes see page Class number $1$ Class group trivial Galois group $QD_{16}$ (as 8T8)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

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

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, -7, 5, 1, 0, 2, -2, 1]);

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

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: $$-107171875$$ -107171875 $$\medspace = -\,5^{6}\cdot 19^{3}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $$10.09$$ 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: $$5$$, $$19$$ 5, 19 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{131}a^{7}-\frac{60}{131}a^{6}-\frac{55}{131}a^{5}+\frac{46}{131}a^{4}-\frac{47}{131}a^{3}-\frac{20}{131}a^{2}-\frac{26}{131}a+\frac{61}{131}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

 Monogenic: Not computed Index: $1$ Inessential primes: None

## 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: $\frac{80}{131}a^{7}-\frac{215}{131}a^{6}+\frac{316}{131}a^{5}-\frac{250}{131}a^{4}+\frac{301}{131}a^{3}+\frac{103}{131}a^{2}-\frac{639}{131}a-\frac{98}{131}$, $\frac{25}{131}a^{7}-\frac{59}{131}a^{6}+\frac{66}{131}a^{5}-\frac{29}{131}a^{4}+\frac{4}{131}a^{3}+\frac{155}{131}a^{2}-\frac{257}{131}a-\frac{47}{131}$, $\frac{28}{131}a^{7}-\frac{108}{131}a^{6}+\frac{163}{131}a^{5}-\frac{153}{131}a^{4}+\frac{125}{131}a^{3}-\frac{36}{131}a^{2}-\frac{335}{131}a+\frac{5}{131}$, $\frac{50}{131}a^{7}-\frac{118}{131}a^{6}+\frac{132}{131}a^{5}-\frac{58}{131}a^{4}+\frac{139}{131}a^{3}+\frac{179}{131}a^{2}-\frac{383}{131}a-\frac{225}{131}$ 80/131*a^7 - 215/131*a^6 + 316/131*a^5 - 250/131*a^4 + 301/131*a^3 + 103/131*a^2 - 639/131*a - 98/131, 25/131*a^7 - 59/131*a^6 + 66/131*a^5 - 29/131*a^4 + 4/131*a^3 + 155/131*a^2 - 257/131*a - 47/131, 28/131*a^7 - 108/131*a^6 + 163/131*a^5 - 153/131*a^4 + 125/131*a^3 - 36/131*a^2 - 335/131*a + 5/131, 50/131*a^7 - 118/131*a^6 + 132/131*a^5 - 58/131*a^4 + 139/131*a^3 + 179/131*a^2 - 383/131*a - 225/131 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$6.79191421448$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{3}\cdot 6.79191421448 \cdot 1}{2\sqrt{107171875}}\approx 0.325477804789$

## Galois group

$\SD_{16}$ (as 8T8):

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 $QD_{16}$ Character table for $QD_{16}$

## Intermediate fields

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

## Sibling fields

 Galois closure: 16.0.4146377695556640625.2

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/padicField/2.8.0.1}{8} }$ ${\href{/padicField/3.8.0.1}{8} }$ R ${\href{/padicField/7.4.0.1}{4} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }$ ${\href{/padicField/17.4.0.1}{4} }^{2}$ R ${\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }$ ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }$ ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
$$5$$ 5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4} 5.4.3.1x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$$19$$ $\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2} 19.2.1.2x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
* 1.5.2t1.a.a$1$ $5$ $$\Q(\sqrt{5})$$ $C_2$ (as 2T1) $1$ $1$
1.19.2t1.a.a$1$ $19$ $$\Q(\sqrt{-19})$$ $C_2$ (as 2T1) $1$ $-1$
1.95.2t1.a.a$1$ $5 \cdot 19$ $$\Q(\sqrt{-95})$$ $C_2$ (as 2T1) $1$ $-1$
* 2.95.4t3.c.a$2$ $5 \cdot 19$ 4.2.475.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.475.8t8.a.a$2$ $5^{2} \cdot 19$ 8.2.107171875.1 $QD_{16}$ (as 8T8) $0$ $0$
* 2.475.8t8.a.b$2$ $5^{2} \cdot 19$ 8.2.107171875.1 $QD_{16}$ (as 8T8) $0$ $0$

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 *.