Properties

Label 8.4.109049934277.1
Degree $8$
Signature $[4, 2]$
Discriminant $37\cdot 233^{4}$
Root discriminant $23.97$
Ramified primes $37, 233$
Class number $1$
Class group Trivial
Galois group $S_4\wr C_2$ (as 8T47)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + x^6 - 7*x^5 + 20*x^4 - 13*x^3 - 46*x^2 + 71*x - 16)
 
gp: K = bnfinit(x^8 - 2*x^7 + x^6 - 7*x^5 + 20*x^4 - 13*x^3 - 46*x^2 + 71*x - 16, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16, 71, -46, -13, 20, -7, 1, -2, 1]);
 

Normalized defining polynomial

\( x^{8} - 2 x^{7} + x^{6} - 7 x^{5} + 20 x^{4} - 13 x^{3} - 46 x^{2} + 71 x - 16 \)

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:  \(109049934277=37\cdot 233^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $23.97$
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:  $37, 233$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{63} a^{7} + \frac{1}{21} a^{6} + \frac{1}{7} a^{5} + \frac{1}{21} a^{4} + \frac{1}{9} a^{3} + \frac{8}{63} a^{2} + \frac{22}{63} a - \frac{8}{63}$

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:  \( 670.841566901 \)
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

\(\Q(\sqrt{233}) \)

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 ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
233Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.37.2t1.a.a$1$ $ 37 $ $x^{2} - x - 9$ $C_2$ (as 2T1) $1$ $1$
1.8621.2t1.a.a$1$ $ 37 \cdot 233 $ $x^{2} - x - 2155$ $C_2$ (as 2T1) $1$ $1$
* 1.233.2t1.a.a$1$ $ 233 $ $x^{2} - x - 58$ $C_2$ (as 2T1) $1$ $1$
2.8621.4t3.c.a$2$ $ 37 \cdot 233 $ $x^{4} - 2 x^{3} + 27 x^{2} - 26 x + 21$ $D_{4}$ (as 4T3) $1$ $-2$
4.17316942353.12t34.b.a$4$ $ 37^{2} \cdot 233^{3}$ $x^{6} - x^{5} + x^{4} - 4 x^{3} + 6 x^{2} + 5 x - 9$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.2749900717.12t34.b.a$4$ $ 37^{3} \cdot 233^{2}$ $x^{6} - x^{5} + x^{4} - 4 x^{3} + 6 x^{2} + 5 x - 9$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.318977.6t13.b.a$4$ $ 37^{2} \cdot 233 $ $x^{6} - x^{5} + x^{4} - 4 x^{3} + 6 x^{2} + 5 x - 9$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.2008693.6t13.b.a$4$ $ 37 \cdot 233^{2}$ $x^{6} - x^{5} + x^{4} - 4 x^{3} + 6 x^{2} + 5 x - 9$ $C_3^2:D_4$ (as 6T13) $1$ $0$
6.23706894081257.12t201.a.a$6$ $ 37^{4} \cdot 233^{3}$ $x^{8} - 2 x^{7} + x^{6} - 7 x^{5} + 20 x^{4} - 13 x^{3} - 46 x^{2} + 71 x - 16$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
6.877155081006509.12t202.a.a$6$ $ 37^{5} \cdot 233^{3}$ $x^{8} - 2 x^{7} + x^{6} - 7 x^{5} + 20 x^{4} - 13 x^{3} - 46 x^{2} + 71 x - 16$ $S_4\wr C_2$ (as 8T47) $1$ $2$
* 6.468025469.8t47.a.a$6$ $ 37 \cdot 233^{3}$ $x^{8} - 2 x^{7} + x^{6} - 7 x^{5} + 20 x^{4} - 13 x^{3} - 46 x^{2} + 71 x - 16$ $S_4\wr C_2$ (as 8T47) $1$ $2$
6.17316942353.12t200.a.a$6$ $ 37^{2} \cdot 233^{3}$ $x^{8} - 2 x^{7} + x^{6} - 7 x^{5} + 20 x^{4} - 13 x^{3} - 46 x^{2} + 71 x - 16$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
9.640726867061.16t1294.a.a$9$ $ 37^{3} \cdot 233^{3}$ $x^{8} - 2 x^{7} + x^{6} - 7 x^{5} + 20 x^{4} - 13 x^{3} - 46 x^{2} + 71 x - 16$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.32454737997240833.18t272.a.a$9$ $ 37^{6} \cdot 233^{3}$ $x^{8} - 2 x^{7} + x^{6} - 7 x^{5} + 20 x^{4} - 13 x^{3} - 46 x^{2} + 71 x - 16$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.410530918173804366777721.18t273.a.a$9$ $ 37^{6} \cdot 233^{6}$ $x^{8} - 2 x^{7} + x^{6} - 7 x^{5} + 20 x^{4} - 13 x^{3} - 46 x^{2} + 71 x - 16$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.8104770066408788557.18t274.a.a$9$ $ 37^{3} \cdot 233^{6}$ $x^{8} - 2 x^{7} + x^{6} - 7 x^{5} + 20 x^{4} - 13 x^{3} - 46 x^{2} + 71 x - 16$ $S_4\wr C_2$ (as 8T47) $1$ $1$
12.15189643972430761570775677.36t1763.a.a$12$ $ 37^{7} \cdot 233^{6}$ $x^{8} - 2 x^{7} + x^{6} - 7 x^{5} + 20 x^{4} - 13 x^{3} - 46 x^{2} + 71 x - 16$ $S_4\wr C_2$ (as 8T47) $1$ $0$
12.11095430220913631534533.24t2821.a.a$12$ $ 37^{5} \cdot 233^{6}$ $x^{8} - 2 x^{7} + x^{6} - 7 x^{5} + 20 x^{4} - 13 x^{3} - 46 x^{2} + 71 x - 16$ $S_4\wr C_2$ (as 8T47) $1$ $0$
18.263038189033177419405010128103547981.36t1758.a.a$18$ $ 37^{9} \cdot 233^{9}$ $x^{8} - 2 x^{7} + x^{6} - 7 x^{5} + 20 x^{4} - 13 x^{3} - 46 x^{2} + 71 x - 16$ $S_4\wr C_2$ (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 *.