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Properties

Label 8.0.5161280.1
Degree $8$
Signature $[0, 4]$
Discriminant $2^{6}\cdot 5\cdot 127^{2}$
Root discriminant $6.90$
Ramified primes $2, 5, 127$
Class number $1$
Class group Trivial
Galois group $C_2 \wr S_4$ (as 8T44)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, 0, 0, -1, 0, 0, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - x^4 + x + 1)
 
gp: K = bnfinit(x^8 - x^7 - x^4 + x + 1, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} - x^{4} + x + 1 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5161280=2^{6}\cdot 5\cdot 127^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $6.90$
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:  $2, 5, 127$
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:  $3$
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:  \( a^{7} - a^{6} - a^{3} + 1 \),  \( a^{7} - 2 a^{6} + 2 a^{5} - 2 a^{4} + 1 \),  \( a^{4} - a^{3} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 0.909386847403 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3:S_4.C_2$ (as 8T44):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 20 conjugacy class representatives for $C_2 \wr S_4$
Character table for $C_2 \wr S_4$

Intermediate fields

4.0.1016.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }$ R ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$127$$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e3_127.2t1.1c1$1$ $ 2^{3} \cdot 127 $ $x^{2} - 254$ $C_2$ (as 2T1) $1$ $1$
1.2e3_5_127.2t1.1c1$1$ $ 2^{3} \cdot 5 \cdot 127 $ $x^{2} - 1270$ $C_2$ (as 2T1) $1$ $1$
1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
2.2e3_127.3t2.1c1$2$ $ 2^{3} \cdot 127 $ $x^{3} - x^{2} - 6 x + 2$ $S_3$ (as 3T2) $1$ $2$
2.2e3_5e2_127.6t3.2c1$2$ $ 2^{3} \cdot 5^{2} \cdot 127 $ $x^{6} - 2 x^{5} - 49 x^{4} + 66 x^{3} + 609 x^{2} - 400 x - 1206$ $D_{6}$ (as 6T3) $1$ $2$
* 3.2e3_127.4t5.1c1$3$ $ 2^{3} \cdot 127 $ $x^{4} - x^{3} + x^{2} - 2 x + 2$ $S_4$ (as 4T5) $1$ $-1$
3.2e6_127e2.6t8.1c1$3$ $ 2^{6} \cdot 127^{2}$ $x^{4} - x^{3} + x^{2} - 2 x + 2$ $S_4$ (as 4T5) $1$ $-1$
3.2e6_5e3_127e2.6t11.2c1$3$ $ 2^{6} \cdot 5^{3} \cdot 127^{2}$ $x^{6} - x^{5} + 11 x^{4} - 18 x^{3} + 22 x^{2} - 112 x + 96$ $S_4\times C_2$ (as 6T11) $1$ $-1$
3.2e3_5e3_127.6t11.2c1$3$ $ 2^{3} \cdot 5^{3} \cdot 127 $ $x^{6} - x^{5} + 11 x^{4} - 18 x^{3} + 22 x^{2} - 112 x + 96$ $S_4\times C_2$ (as 6T11) $1$ $-1$
* 4.2e3_5_127.8t44.1c1$4$ $ 2^{3} \cdot 5 \cdot 127 $ $x^{8} - x^{7} - x^{4} + x + 1$ $C_2 \wr S_4$ (as 8T44) $1$ $0$
4.2e3_5e3_127.8t44.1c1$4$ $ 2^{3} \cdot 5^{3} \cdot 127 $ $x^{8} - x^{7} - x^{4} + x + 1$ $C_2 \wr S_4$ (as 8T44) $1$ $0$
4.2e9_5_127e3.8t44.1c1$4$ $ 2^{9} \cdot 5 \cdot 127^{3}$ $x^{8} - x^{7} - x^{4} + x + 1$ $C_2 \wr S_4$ (as 8T44) $1$ $0$
4.2e9_5e3_127e3.8t44.1c1$4$ $ 2^{9} \cdot 5^{3} \cdot 127^{3}$ $x^{8} - x^{7} - x^{4} + x + 1$ $C_2 \wr S_4$ (as 8T44) $1$ $0$
6.2e6_5e3_127e2.8t41.1c1$6$ $ 2^{6} \cdot 5^{3} \cdot 127^{2}$ $x^{8} - 3 x^{7} + 6 x^{5} - 6 x^{4} + 7 x^{3} - 8 x^{2} - 2 x + 1$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $2$
6.2e12_5e3_127e4.12t111.1c1$6$ $ 2^{12} \cdot 5^{3} \cdot 127^{4}$ $x^{8} - 3 x^{7} + 6 x^{5} - 6 x^{4} + 7 x^{3} - 8 x^{2} - 2 x + 1$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $2$
6.2e9_5e3_127e3.8t41.1c1$6$ $ 2^{9} \cdot 5^{3} \cdot 127^{3}$ $x^{8} - 3 x^{7} + 6 x^{5} - 6 x^{4} + 7 x^{3} - 8 x^{2} - 2 x + 1$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $-2$
6.2e9_5e3_127e3.12t108.1c1$6$ $ 2^{9} \cdot 5^{3} \cdot 127^{3}$ $x^{8} - 3 x^{7} + 6 x^{5} - 6 x^{4} + 7 x^{3} - 8 x^{2} - 2 x + 1$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $-2$
8.2e12_5e6_127e4.24t708.1c1$8$ $ 2^{12} \cdot 5^{6} \cdot 127^{4}$ $x^{8} - x^{7} - x^{4} + x + 1$ $C_2 \wr S_4$ (as 8T44) $1$ $0$
8.2e12_5e2_127e4.24t1151.1c1$8$ $ 2^{12} \cdot 5^{2} \cdot 127^{4}$ $x^{8} - x^{7} - x^{4} + x + 1$ $C_2 \wr S_4$ (as 8T44) $1$ $0$

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 *.