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Properties

Label 8.0.1406080000.1
Degree $8$
Signature $[0, 4]$
Discriminant $2^{10}\cdot 5^{4}\cdot 13^{3}$
Root discriminant $13.92$
Ramified primes $2, 5, 13$
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![13, 0, 0, 0, 2, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^6 + 2*x^4 + 13)
 
gp: K = bnfinit(x^8 - 4*x^6 + 2*x^4 + 13, 1)
 

Normalized defining polynomial

\( x^{8} - 4 x^{6} + 2 x^{4} + 13 \)

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:  \(1406080000=2^{10}\cdot 5^{4}\cdot 13^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.92$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{4}$

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:  \( \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{3}{2} \),  \( \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} + a^{4} - \frac{5}{4} a^{3} + a^{2} - \frac{3}{4} a + \frac{3}{2} \),  \( \frac{1}{2} a^{4} - 3 a^{2} + \frac{9}{2} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 65.2516884581 \)
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

4.0.5200.1

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 R ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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

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$2.8.10.3$x^{8} + 4 x^{2} + 20$$8$$1$$10$$\textrm{GL(2,3)}$$[4/3, 4/3, 3/2]_{3}^{2}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
$13$13.4.3.4$x^{4} + 104$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.13.2t1.1c1$1$ $ 13 $ $x^{2} - x - 3$ $C_2$ (as 2T1) $1$ $1$
2.2e2_5e2_13.3t2.2c1$2$ $ 2^{2} \cdot 5^{2} \cdot 13 $ $x^{3} - 10 x - 10$ $S_3$ (as 3T2) $1$ $2$
3.2e4_5e2_13e2.6t8.1c1$3$ $ 2^{4} \cdot 5^{2} \cdot 13^{2}$ $x^{4} + 4 x^{2} - 2 x + 2$ $S_4$ (as 4T5) $1$ $-1$
3.2e2_5e2_13e3.4t5.1c1$3$ $ 2^{2} \cdot 5^{2} \cdot 13^{3}$ $x^{4} - 2 x^{3} + 8 x^{2} + 6 x + 35$ $S_4$ (as 4T5) $1$ $-1$
3.2e4_5e2_13e2.6t8.2c1$3$ $ 2^{4} \cdot 5^{2} \cdot 13^{2}$ $x^{4} + 26 x^{2} - 26 x + 117$ $S_4$ (as 4T5) $1$ $-1$
3.2e4_5e2_13e3.4t5.2c1$3$ $ 2^{4} \cdot 5^{2} \cdot 13^{3}$ $x^{4} + 26 x^{2} - 26 x + 117$ $S_4$ (as 4T5) $1$ $-1$
3.2e2_5e2_13e2.6t8.1c1$3$ $ 2^{2} \cdot 5^{2} \cdot 13^{2}$ $x^{4} - 2 x^{3} + 8 x^{2} + 6 x + 35$ $S_4$ (as 4T5) $1$ $-1$
* 3.2e4_5e2_13.4t5.1c1$3$ $ 2^{4} \cdot 5^{2} \cdot 13 $ $x^{4} + 4 x^{2} - 2 x + 2$ $S_4$ (as 4T5) $1$ $-1$
4.2e6_5e2_13e4.8t40.1c1$4$ $ 2^{6} \cdot 5^{2} \cdot 13^{4}$ $x^{8} - 4 x^{6} + 2 x^{4} + 13$ $Q_8:S_4$ (as 8T40) $1$ $0$
* 4.2e6_5e2_13e2.8t40.1c1$4$ $ 2^{6} \cdot 5^{2} \cdot 13^{2}$ $x^{8} - 4 x^{6} + 2 x^{4} + 13$ $Q_8:S_4$ (as 8T40) $1$ $0$
6.2e8_5e4_13e5.8t34.1c1$6$ $ 2^{8} \cdot 5^{4} \cdot 13^{5}$ $x^{8} - 4 x^{7} + 14 x^{5} - 49 x^{4} + 18 x^{3} + 20 x + 27$ $V_4^2:S_3$ (as 8T34) $1$ $2$
8.2e12_5e6_13e6.24t332.1c1$8$ $ 2^{12} \cdot 5^{6} \cdot 13^{6}$ $x^{8} - 4 x^{6} + 2 x^{4} + 13$ $Q_8:S_4$ (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 *.