Properties

Label 8.4.6529513515605.1
Degree $8$
Signature $[4, 2]$
Discriminant $5\cdot 1069^{4}$
Root discriminant $39.98$
Ramified primes $5, 1069$
Class number $1$
Class group Trivial
Galois group $S_4\wr C_2$ (as 8T47)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, 56, 108, -178, 87, -39, 11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + 11*x^6 - 39*x^5 + 87*x^4 - 178*x^3 + 108*x^2 + 56*x + 5)
 
gp: K = bnfinit(x^8 - 2*x^7 + 11*x^6 - 39*x^5 + 87*x^4 - 178*x^3 + 108*x^2 + 56*x + 5, 1)
 

Normalized defining polynomial

\( x^{8} - 2 x^{7} + 11 x^{6} - 39 x^{5} + 87 x^{4} - 178 x^{3} + 108 x^{2} + 56 x + 5 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6529513515605=5\cdot 1069^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.98$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 1069$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{9} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{5985} a^{7} + \frac{307}{5985} a^{6} + \frac{148}{1995} a^{5} - \frac{277}{665} a^{4} + \frac{121}{399} a^{3} - \frac{1933}{5985} a^{2} - \frac{47}{105} a - \frac{232}{1197}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $5$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 5071.00887402 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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

\(\Q(\sqrt{1069}) \)

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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
1069Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.5_1069.2t1.1c1$1$ $ 5 \cdot 1069 $ $x^{2} - x - 1336$ $C_2$ (as 2T1) $1$ $1$
* 1.1069.2t1.1c1$1$ $ 1069 $ $x^{2} - x - 267$ $C_2$ (as 2T1) $1$ $1$
2.5_1069.4t3.2c1$2$ $ 5 \cdot 1069 $ $x^{4} - 2 x^{3} + 18 x^{2} - 17 x + 71$ $D_{4}$ (as 4T3) $1$ $-2$
4.5e2_1069.6t13.1c1$4$ $ 5^{2} \cdot 1069 $ $x^{6} + 2 x^{4} - x^{3} + x^{2} - x - 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.5e3_1069e2.12t34.1c1$4$ $ 5^{3} \cdot 1069^{2}$ $x^{6} + 2 x^{4} - x^{3} + x^{2} - x - 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.5e2_1069e3.12t34.1c1$4$ $ 5^{2} \cdot 1069^{3}$ $x^{6} + 2 x^{4} - x^{3} + x^{2} - x - 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.5_1069e2.6t13.1c1$4$ $ 5 \cdot 1069^{2}$ $x^{6} + 2 x^{4} - x^{3} + x^{2} - x - 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
6.5e4_1069e3.12t201.1c1$6$ $ 5^{4} \cdot 1069^{3}$ $x^{8} - 2 x^{7} + 11 x^{6} - 39 x^{5} + 87 x^{4} - 178 x^{3} + 108 x^{2} + 56 x + 5$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
6.5e5_1069e3.12t202.1c1$6$ $ 5^{5} \cdot 1069^{3}$ $x^{8} - 2 x^{7} + 11 x^{6} - 39 x^{5} + 87 x^{4} - 178 x^{3} + 108 x^{2} + 56 x + 5$ $S_4\wr C_2$ (as 8T47) $1$ $2$
* 6.5_1069e3.8t47.1c1$6$ $ 5 \cdot 1069^{3}$ $x^{8} - 2 x^{7} + 11 x^{6} - 39 x^{5} + 87 x^{4} - 178 x^{3} + 108 x^{2} + 56 x + 5$ $S_4\wr C_2$ (as 8T47) $1$ $2$
6.5e2_1069e3.12t200.1c1$6$ $ 5^{2} \cdot 1069^{3}$ $x^{8} - 2 x^{7} + 11 x^{6} - 39 x^{5} + 87 x^{4} - 178 x^{3} + 108 x^{2} + 56 x + 5$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
9.5e3_1069e3.16t1294.1c1$9$ $ 5^{3} \cdot 1069^{3}$ $x^{8} - 2 x^{7} + 11 x^{6} - 39 x^{5} + 87 x^{4} - 178 x^{3} + 108 x^{2} + 56 x + 5$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.5e6_1069e3.18t272.1c1$9$ $ 5^{6} \cdot 1069^{3}$ $x^{8} - 2 x^{7} + 11 x^{6} - 39 x^{5} + 87 x^{4} - 178 x^{3} + 108 x^{2} + 56 x + 5$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.5e6_1069e6.18t273.1c1$9$ $ 5^{6} \cdot 1069^{6}$ $x^{8} - 2 x^{7} + 11 x^{6} - 39 x^{5} + 87 x^{4} - 178 x^{3} + 108 x^{2} + 56 x + 5$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.5e3_1069e6.18t274.1c1$9$ $ 5^{3} \cdot 1069^{6}$ $x^{8} - 2 x^{7} + 11 x^{6} - 39 x^{5} + 87 x^{4} - 178 x^{3} + 108 x^{2} + 56 x + 5$ $S_4\wr C_2$ (as 8T47) $1$ $1$
12.5e7_1069e6.36t1944.1c1$12$ $ 5^{7} \cdot 1069^{6}$ $x^{8} - 2 x^{7} + 11 x^{6} - 39 x^{5} + 87 x^{4} - 178 x^{3} + 108 x^{2} + 56 x + 5$ $S_4\wr C_2$ (as 8T47) $1$ $0$
12.5e5_1069e6.24t2821.1c1$12$ $ 5^{5} \cdot 1069^{6}$ $x^{8} - 2 x^{7} + 11 x^{6} - 39 x^{5} + 87 x^{4} - 178 x^{3} + 108 x^{2} + 56 x + 5$ $S_4\wr C_2$ (as 8T47) $1$ $0$
18.5e9_1069e9.36t1758.1c1$18$ $ 5^{9} \cdot 1069^{9}$ $x^{8} - 2 x^{7} + 11 x^{6} - 39 x^{5} + 87 x^{4} - 178 x^{3} + 108 x^{2} + 56 x + 5$ $S_4\wr C_2$ (as 8T47) $1$ $-2$

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 *.