# Properties

 Label 8.0.8719893.2 Degree $8$ Signature $[0, 4]$ Discriminant $8719893$ Root discriminant $$7.37$$ Ramified primes see page Class number $1$ Class group trivial Galois group $C_4\wr C_2$ (as 8T17)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

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

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, -2, -5, -3, 1, 5, 0, -3, 1]);

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

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $8$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 4]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$8719893$$ 8719893 $$\medspace = 3^{4}\cdot 7^{2}\cdot 13^{3}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $$7.37$$ 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: $$3$$, $$7$$, $$13$$ 3, 7, 13 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $\card{ \Aut(K/\Q) }$: $4$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

 Monogenic: Yes 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: $3$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$4 a^{7} - 6 a^{6} - 9 a^{5} + 6 a^{4} + 14 a^{3} + 9 a^{2} - 7 a - 18$$ 4*a^(7) - 6*a^(6) - 9*a^(5) + 6*a^(4) + 14*a^(3) + 9*a^(2) - 7*a - 18  (order $6$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $a^{7}-a^{6}-2a^{5}+2a^{3}+3a^{2}-3$, $3a^{7}-4a^{6}-7a^{5}+4a^{4}+10a^{3}+6a^{2}-4a-12$, $a-1$ a^7 - a^6 - 2*a^5 + 2*a^3 + 3*a^2 - 3, 3*a^7 - 4*a^6 - 7*a^5 + 4*a^4 + 10*a^3 + 6*a^2 - 4*a - 12, a - 1 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$3.59771330101$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{4}\cdot 3.59771330101 \cdot 1}{6\sqrt{8719893}}\approx 0.316474845801$

## Galois group

$C_4\wr C_2$ (as 8T17):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 32 The 14 conjugacy class representatives for $C_4\wr C_2$ Character table for $C_4\wr C_2$

## Intermediate fields

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

## Sibling fields

 Galois closure: data not computed Degree 8 sibling: data not computed Degree 16 siblings: data not computed

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/padicField/2.8.0.1}{8} }$ R ${\href{/padicField/5.8.0.1}{8} }$ R ${\href{/padicField/11.8.0.1}{8} }$ R ${\href{/padicField/17.2.0.1}{2} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }$ ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.8.0.1}{8} }$ ${\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }$

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
$$3$$ 3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4} $$7$$ \Q_{7}$$x + 2$$1$$1$$0Trivial[\ ] \Q_{7}$$x + 2$$1$$1$$0Trivial[\ ] \Q_{7}$$x + 2$$1$$1$$0Trivial[\ ] \Q_{7}$$x + 2$$1$$1$$0Trivial[\ ] 7.4.2.2x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$$13$$ 13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4} 13.4.3.4x^{4} + 104$$4$$1$$3$$C_4$$[\ ]_{4}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
1.13.2t1.a.a$1$ $13$ $$\Q(\sqrt{13})$$ $C_2$ (as 2T1) $1$ $1$
1.39.2t1.a.a$1$ $3 \cdot 13$ $$\Q(\sqrt{-39})$$ $C_2$ (as 2T1) $1$ $-1$
* 1.3.2t1.a.a$1$ $3$ $$\Q(\sqrt{-3})$$ $C_2$ (as 2T1) $1$ $-1$
1.91.4t1.a.a$1$ $7 \cdot 13$ 4.4.107653.1 $C_4$ (as 4T1) $0$ $1$
1.91.4t1.a.b$1$ $7 \cdot 13$ 4.4.107653.1 $C_4$ (as 4T1) $0$ $1$
1.273.4t1.a.a$1$ $3 \cdot 7 \cdot 13$ 4.0.968877.2 $C_4$ (as 4T1) $0$ $-1$
1.273.4t1.a.b$1$ $3 \cdot 7 \cdot 13$ 4.0.968877.2 $C_4$ (as 4T1) $0$ $-1$
2.24843.4t3.a.a$2$ $3 \cdot 7^{2} \cdot 13^{2}$ 4.0.968877.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.39.4t3.b.a$2$ $3 \cdot 13$ 4.2.507.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.273.8t17.d.a$2$ $3 \cdot 7 \cdot 13$ 8.0.8719893.2 $C_4\wr C_2$ (as 8T17) $0$ $0$
2.3549.8t17.d.a$2$ $3 \cdot 7 \cdot 13^{2}$ 8.0.8719893.2 $C_4\wr C_2$ (as 8T17) $0$ $0$
* 2.273.8t17.d.b$2$ $3 \cdot 7 \cdot 13$ 8.0.8719893.2 $C_4\wr C_2$ (as 8T17) $0$ $0$
2.3549.8t17.d.b$2$ $3 \cdot 7 \cdot 13^{2}$ 8.0.8719893.2 $C_4\wr C_2$ (as 8T17) $0$ $0$

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 *.