# Properties

 Label 8.4.9191328125.1 Degree $8$ Signature $[4, 2]$ Discriminant $9191328125$ Root discriminant $17.60$ Ramified primes $5, 7$ Class number $1$ Class group trivial Galois group $C_8:C_2$ (as 8T7)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 13*x^6 - 13*x^5 + 25*x^4 + 38*x^3 - 33*x^2 - 34*x + 11)

gp: K = bnfinit(x^8 - x^7 - 13*x^6 - 13*x^5 + 25*x^4 + 38*x^3 - 33*x^2 - 34*x + 11, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, -34, -33, 38, 25, -13, -13, -1, 1]);

$$x^{8} - x^{7} - 13 x^{6} - 13 x^{5} + 25 x^{4} + 38 x^{3} - 33 x^{2} - 34 x + 11$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $8$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[4, 2]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$9191328125$$$$\medspace = 5^{7}\cdot 7^{6}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $17.60$ 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: $5, 7$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $4$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{59851} a^{7} + \frac{2025}{59851} a^{6} - \frac{2462}{5441} a^{5} + \frac{15222}{59851} a^{4} + \frac{16532}{59851} a^{3} - \frac{22690}{59851} a^{2} - \frac{4405}{59851} a - \frac{615}{5441}$

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: $5$ 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: $$129.505077804$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{4}\cdot(2\pi)^{2}\cdot 129.505077804 \cdot 1}{2\sqrt{9191328125}}\approx 0.426626065191$

## Galois group

$OD_{16}$ (as 8T7):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 16 The 10 conjugacy class representatives for $C_8:C_2$ Character table for $C_8:C_2$

## Intermediate fields

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

## Sibling fields

 Galois closure: 16.0.84480512701416015625.1

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }$ ${\href{/LocalNumberField/3.8.0.1}{8} }$ R R ${\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2} 77.8.6.3x^{8} - 7 x^{4} + 147$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
1.35.2t1.a.a$1$ $5 \cdot 7$ $$\Q(\sqrt{-35})$$ $C_2$ (as 2T1) $1$ $-1$
1.7.2t1.a.a$1$ $7$ $$\Q(\sqrt{-7})$$ $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $5$ $$\Q(\sqrt{5})$$ $C_2$ (as 2T1) $1$ $1$
* 1.35.4t1.a.a$1$ $5 \cdot 7$ 4.4.6125.1 $C_4$ (as 4T1) $0$ $1$
1.5.4t1.a.a$1$ $5$ $$\Q(\zeta_{5})$$ $C_4$ (as 4T1) $0$ $-1$
1.5.4t1.a.b$1$ $5$ $$\Q(\zeta_{5})$$ $C_4$ (as 4T1) $0$ $-1$
* 1.35.4t1.a.b$1$ $5 \cdot 7$ 4.4.6125.1 $C_4$ (as 4T1) $0$ $1$
* 2.1225.8t7.a.a$2$ $5^{2} \cdot 7^{2}$ 8.4.9191328125.1 $C_8:C_2$ (as 8T7) $0$ $0$
* 2.1225.8t7.a.b$2$ $5^{2} \cdot 7^{2}$ 8.4.9191328125.1 $C_8:C_2$ (as 8T7) $0$ $0$

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 *.