# Properties

 Label 8.6.321078613651227.1 Degree $8$ Signature $[6, 1]$ Discriminant $-\,3^{7}\cdot 619^{4}$ Root discriminant $65.06$ Ramified primes $3, 619$ Class number $1$ Class group Trivial Galois group $S_4\wr C_2$ (as 8T47)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 - 14*x^6 + 17*x^5 + 43*x^4 + 323*x^3 + 310*x^2 - 1135*x - 413)

gp: K = bnfinit(x^8 - 4*x^7 - 14*x^6 + 17*x^5 + 43*x^4 + 323*x^3 + 310*x^2 - 1135*x - 413, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-413, -1135, 310, 323, 43, 17, -14, -4, 1]);

## Normalizeddefining polynomial

$$x^{8} - 4 x^{7} - 14 x^{6} + 17 x^{5} + 43 x^{4} + 323 x^{3} + 310 x^{2} - 1135 x - 413$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $8$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[6, 1]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-321078613651227=-\,3^{7}\cdot 619^{4}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $65.06$ 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: $3, 619$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $1$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{14371412} a^{7} - \frac{136173}{3592853} a^{6} + \frac{173901}{14371412} a^{5} + \frac{1890737}{7185706} a^{4} - \frac{5556523}{14371412} a^{3} - \frac{4852817}{14371412} a^{2} + \frac{5641453}{14371412} a - \frac{1110313}{14371412}$

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: $6$ 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: $$100453.738914$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$S_4\wr C_2$ (as 8T47):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 1152 The 20 conjugacy class representatives for $S_4\wr C_2$ Character table for $S_4\wr C_2$

## Intermediate fields

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

## Sibling fields

 Degree 12 siblings: data not computed Degree 16 siblings: data not computed Degree 18 siblings: data not computed Degree 24 siblings: data not computed Degree 32 siblings: data not computed Degree 36 siblings: data not computed

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ R ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$3.8.7.2$x^{8} - 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
619Data not computed

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.3.2t1.a.a$1$ $3$ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.619.2t1.a.a$1$ $619$ $x^{2} - x + 155$ $C_2$ (as 2T1) $1$ $-1$
* 1.1857.2t1.a.a$1$ $3 \cdot 619$ $x^{2} - x - 464$ $C_2$ (as 2T1) $1$ $1$
2.5571.4t3.c.a$2$ $3^{2} \cdot 619$ $x^{4} - x^{3} + 21 x^{2} - 4 x + 112$ $D_{4}$ (as 4T3) $1$ $0$
4.16713.6t13.b.a$4$ $3^{3} \cdot 619$ $x^{6} - x^{5} + 2 x^{4} - x^{3} - x^{2} + 2 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.10345347.12t34.b.a$4$ $3^{3} \cdot 619^{2}$ $x^{6} - x^{5} + 2 x^{4} - x^{3} - x^{2} + 2 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $-2$
4.6403769793.12t34.b.a$4$ $3^{3} \cdot 619^{3}$ $x^{6} - x^{5} + 2 x^{4} - x^{3} - x^{2} + 2 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.10345347.6t13.b.a$4$ $3^{3} \cdot 619^{2}$ $x^{6} - x^{5} + 2 x^{4} - x^{3} - x^{2} + 2 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $2$
6.57633928137.12t201.a.a$6$ $3^{5} \cdot 619^{3}$ $x^{8} - 4 x^{7} - 14 x^{6} + 17 x^{5} + 43 x^{4} + 323 x^{3} + 310 x^{2} - 1135 x - 413$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
6.172901784411.12t202.a.a$6$ $3^{6} \cdot 619^{3}$ $x^{8} - 4 x^{7} - 14 x^{6} + 17 x^{5} + 43 x^{4} + 323 x^{3} + 310 x^{2} - 1135 x - 413$ $S_4\wr C_2$ (as 8T47) $1$ $-4$
* 6.172901784411.8t47.a.a$6$ $3^{6} \cdot 619^{3}$ $x^{8} - 4 x^{7} - 14 x^{6} + 17 x^{5} + 43 x^{4} + 323 x^{3} + 310 x^{2} - 1135 x - 413$ $S_4\wr C_2$ (as 8T47) $1$ $4$
6.57633928137.12t200.a.a$6$ $3^{5} \cdot 619^{3}$ $x^{8} - 4 x^{7} - 14 x^{6} + 17 x^{5} + 43 x^{4} + 323 x^{3} + 310 x^{2} - 1135 x - 413$ $S_4\wr C_2$ (as 8T47) $1$ $2$
9.1556116059699.16t1294.a.a$9$ $3^{8} \cdot 619^{3}$ $x^{8} - 4 x^{7} - 14 x^{6} + 17 x^{5} + 43 x^{4} + 323 x^{3} + 310 x^{2} - 1135 x - 413$ $S_4\wr C_2$ (as 8T47) $1$ $3$
9.518705353233.18t272.a.a$9$ $3^{7} \cdot 619^{3}$ $x^{8} - 4 x^{7} - 14 x^{6} + 17 x^{5} + 43 x^{4} + 323 x^{3} + 310 x^{2} - 1135 x - 413$ $S_4\wr C_2$ (as 8T47) $1$ $-3$
9.369074408055653365641.18t273.a.a$9$ $3^{8} \cdot 619^{6}$ $x^{8} - 4 x^{7} - 14 x^{6} + 17 x^{5} + 43 x^{4} + 323 x^{3} + 310 x^{2} - 1135 x - 413$ $S_4\wr C_2$ (as 8T47) $1$ $-3$
9.123024802685217788547.18t274.a.a$9$ $3^{7} \cdot 619^{6}$ $x^{8} - 4 x^{7} - 14 x^{6} + 17 x^{5} + 43 x^{4} + 323 x^{3} + 310 x^{2} - 1135 x - 413$ $S_4\wr C_2$ (as 8T47) $1$ $3$
12.9965009017502640872307.36t1763.a.a$12$ $3^{11} \cdot 619^{6}$ $x^{8} - 4 x^{7} - 14 x^{6} + 17 x^{5} + 43 x^{4} + 323 x^{3} + 310 x^{2} - 1135 x - 413$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
12.9965009017502640872307.24t2821.a.a$12$ $3^{11} \cdot 619^{6}$ $x^{8} - 4 x^{7} - 14 x^{6} + 17 x^{5} + 43 x^{4} + 323 x^{3} + 310 x^{2} - 1135 x - 413$ $S_4\wr C_2$ (as 8T47) $1$ $2$
18.574322613599304179242260631402059.36t1758.a.a$18$ $3^{16} \cdot 619^{9}$ $x^{8} - 4 x^{7} - 14 x^{6} + 17 x^{5} + 43 x^{4} + 323 x^{3} + 310 x^{2} - 1135 x - 413$ $S_4\wr C_2$ (as 8T47) $1$ $0$

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 *.