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Properties

Label 8.4.32993536.1
Degree $8$
Signature $[4, 2]$
Discriminant $2^{8}\cdot 359^{2}$
Root discriminant $8.71$
Ramified primes $2, 359$
Class number $1$
Class group Trivial
Galois group $C_2^3:S_4$ (as 8T39)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 4, -3, -6, 14, -16, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 10*x^6 - 16*x^5 + 14*x^4 - 6*x^3 - 3*x^2 + 4*x - 1)
 
gp: K = bnfinit(x^8 - 4*x^7 + 10*x^6 - 16*x^5 + 14*x^4 - 6*x^3 - 3*x^2 + 4*x - 1, 1)
 

Normalized defining polynomial

\( x^{8} - 4 x^{7} + 10 x^{6} - 16 x^{5} + 14 x^{4} - 6 x^{3} - 3 x^{2} + 4 x - 1 \)

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:  \(32993536=2^{8}\cdot 359^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $8.71$
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:  $2, 359$
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 \(a\))

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

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:  \( a \),  \( a^{7} - 3 a^{6} + 7 a^{5} - 9 a^{4} + 5 a^{3} - a^{2} - 4 a \),  \( 10 a^{7} - 35 a^{6} + 83 a^{5} - 120 a^{4} + 83 a^{3} - 22 a^{2} - 40 a + 20 \),  \( 11 a^{7} - 39 a^{6} + 92 a^{5} - 133 a^{4} + 91 a^{3} - 21 a^{2} - 45 a + 23 \),  \( 8 a^{7} - 28 a^{6} + 66 a^{5} - 95 a^{4} + 64 a^{3} - 15 a^{2} - 34 a + 17 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3.76255994435 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3:S_4$ (as 8T39):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 192
The 13 conjugacy class representatives for $C_2^3:S_4$
Character table for $C_2^3:S_4$

Intermediate fields

4.4.5744.1

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

Sibling fields

Degree 8 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.8.8.2$x^{8} + 2 x^{7} + 8 x^{2} + 48$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{4}$
359Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e2_359.2t1.1c1$1$ $ 2^{2} \cdot 359 $ $x^{2} - 359$ $C_2$ (as 2T1) $1$ $1$
2.2e2_359.3t2.1c1$2$ $ 2^{2} \cdot 359 $ $x^{3} - 11 x - 12$ $S_3$ (as 3T2) $1$ $2$
3.2e2_359.4t5.1c1$3$ $ 2^{2} \cdot 359 $ $x^{4} - x^{3} + 3 x^{2} + 2$ $S_4$ (as 4T5) $1$ $-1$
3.2e6_359e2.6t8.2c1$3$ $ 2^{6} \cdot 359^{2}$ $x^{4} + x^{2} - 4 x + 3$ $S_4$ (as 4T5) $1$ $-1$
* 3.2e4_359.4t5.1c1$3$ $ 2^{4} \cdot 359 $ $x^{4} - 5 x^{2} - 2 x + 1$ $S_4$ (as 4T5) $1$ $3$
3.2e6_359e2.6t8.1c1$3$ $ 2^{6} \cdot 359^{2}$ $x^{4} - 5 x^{2} - 2 x + 1$ $S_4$ (as 4T5) $1$ $3$
3.2e4_359.4t5.2c1$3$ $ 2^{4} \cdot 359 $ $x^{4} + x^{2} - 4 x + 3$ $S_4$ (as 4T5) $1$ $-1$
3.2e4_359e2.6t8.1c1$3$ $ 2^{4} \cdot 359^{2}$ $x^{4} - x^{3} + 3 x^{2} + 2$ $S_4$ (as 4T5) $1$ $-1$
* 4.2e4_359.8t39.1c1$4$ $ 2^{4} \cdot 359 $ $x^{8} - 4 x^{7} + 10 x^{6} - 16 x^{5} + 14 x^{4} - 6 x^{3} - 3 x^{2} + 4 x - 1$ $C_2^3:S_4$ (as 8T39) $1$ $0$
4.2e8_359e3.8t39.1c1$4$ $ 2^{8} \cdot 359^{3}$ $x^{8} - 4 x^{7} + 10 x^{6} - 16 x^{5} + 14 x^{4} - 6 x^{3} - 3 x^{2} + 4 x - 1$ $C_2^3:S_4$ (as 8T39) $1$ $0$
6.2e10_359e3.8t34.1c1$6$ $ 2^{10} \cdot 359^{3}$ $x^{8} + 8 x^{6} - 24 x^{5} + 118 x^{4} - 96 x^{3} + 552 x^{2} - 1224 x + 1165$ $V_4^2:S_3$ (as 8T34) $1$ $-2$
8.2e12_359e4.24t333.1c1$8$ $ 2^{12} \cdot 359^{4}$ $x^{8} - 4 x^{7} + 10 x^{6} - 16 x^{5} + 14 x^{4} - 6 x^{3} - 3 x^{2} + 4 x - 1$ $C_2^3:S_4$ (as 8T39) $1$ $0$

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 *.