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Properties

Label 8.0.338608873.1
Degree $8$
Signature $[0, 4]$
Discriminant $17^{3}\cdot 41^{3}$
Root discriminant $11.65$
Ramified primes $17, 41$
Class number $1$
Class group Trivial
Galois group $Q_8:S_4$ (as 8T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, -4, 2, -12, 6, 4, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 + 4*x^5 + 6*x^4 - 12*x^3 + 2*x^2 - 4*x + 7)
 
gp: K = bnfinit(x^8 - 3*x^7 + 4*x^5 + 6*x^4 - 12*x^3 + 2*x^2 - 4*x + 7, 1)
 

Normalized defining polynomial

\( x^{8} - 3 x^{7} + 4 x^{5} + 6 x^{4} - 12 x^{3} + 2 x^{2} - 4 x + 7 \)

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:  \(338608873=17^{3}\cdot 41^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.65$
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:  $17, 41$
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:  \( 3 a^{7} - 5 a^{6} - 7 a^{5} + 3 a^{4} + 23 a^{3} - 6 a^{2} - 3 a - 16 \),  \( a - 1 \),  \( a^{6} - 2 a^{4} - 3 a^{3} + 2 a^{2} + 2 a + 3 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 9.58951414655 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3:A_4:C_2$ (as 8T40):

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 $Q_8:S_4$
Character table for $Q_8:S_4$

Intermediate fields

4.0.697.1

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ R ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.17_41.2t1.1c1$1$ $ 17 \cdot 41 $ $x^{2} - x - 174$ $C_2$ (as 2T1) $1$ $1$
2.17_41.3t2.1c1$2$ $ 17 \cdot 41 $ $x^{3} - 7 x - 5$ $S_3$ (as 3T2) $1$ $2$
3.17e2_41e2.6t8.2c1$3$ $ 17^{2} \cdot 41^{2}$ $x^{4} - x^{3} + 2 x^{2} - x + 2$ $S_4$ (as 4T5) $1$ $-1$
3.17e3_41.4t5.1c1$3$ $ 17^{3} \cdot 41 $ $x^{4} - x^{3} - 6 x^{2} + x + 18$ $S_4$ (as 4T5) $1$ $-1$
3.17e2_41e2.6t8.4c1$3$ $ 17^{2} \cdot 41^{2}$ $x^{4} - x^{3} + 11 x^{2} + 18 x + 35$ $S_4$ (as 4T5) $1$ $-1$
3.17e3_41.4t5.2c1$3$ $ 17^{3} \cdot 41 $ $x^{4} - x^{3} + 11 x^{2} + 18 x + 35$ $S_4$ (as 4T5) $1$ $-1$
* 3.17_41.4t5.1c1$3$ $ 17 \cdot 41 $ $x^{4} - x^{3} + 2 x^{2} - x + 2$ $S_4$ (as 4T5) $1$ $-1$
3.17e2_41e2.6t8.3c1$3$ $ 17^{2} \cdot 41^{2}$ $x^{4} - x^{3} - 6 x^{2} + x + 18$ $S_4$ (as 4T5) $1$ $-1$
4.17e4_41e2.8t40.2c1$4$ $ 17^{4} \cdot 41^{2}$ $x^{8} - 3 x^{7} + 4 x^{5} + 6 x^{4} - 12 x^{3} + 2 x^{2} - 4 x + 7$ $Q_8:S_4$ (as 8T40) $1$ $0$
* 4.17e2_41e2.8t40.2c1$4$ $ 17^{2} \cdot 41^{2}$ $x^{8} - 3 x^{7} + 4 x^{5} + 6 x^{4} - 12 x^{3} + 2 x^{2} - 4 x + 7$ $Q_8:S_4$ (as 8T40) $1$ $0$
6.17e5_41e3.8t34.1c1$6$ $ 17^{5} \cdot 41^{3}$ $x^{8} - 2 x^{7} + 40 x^{5} - 226 x^{4} + 342 x^{3} - 136 x^{2} + 95 x - 132$ $V_4^2:S_3$ (as 8T34) $1$ $2$
8.17e6_41e4.24t332.2c1$8$ $ 17^{6} \cdot 41^{4}$ $x^{8} - 3 x^{7} + 4 x^{5} + 6 x^{4} - 12 x^{3} + 2 x^{2} - 4 x + 7$ $Q_8:S_4$ (as 8T40) $1$ $0$

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 *.