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Properties

Label 8.0.57760000.3
Degree $8$
Signature $[0, 4]$
Discriminant $2^{8}\cdot 5^{4}\cdot 19^{2}$
Root discriminant $9.34$
Ramified primes $2, 5, 19$
Class number $1$
Class group Trivial
Galois group $A_4\wr C_2$ (as 8T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 10, -12, 7, -6, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 4*x^6 - 6*x^5 + 7*x^4 - 12*x^3 + 10*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^8 + 4*x^6 - 6*x^5 + 7*x^4 - 12*x^3 + 10*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{8} + 4 x^{6} - 6 x^{5} + 7 x^{4} - 12 x^{3} + 10 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(57760000=2^{8}\cdot 5^{4}\cdot 19^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $9.34$
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:  $2, 5, 19$
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 \(a\))

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

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:  \( \frac{8}{9} a^{7} + \frac{4}{9} a^{6} + \frac{34}{9} a^{5} - \frac{31}{9} a^{4} + 4 a^{3} - \frac{26}{3} a^{2} + \frac{32}{9} a - \frac{7}{9} \),  \( \frac{4}{9} a^{7} + \frac{2}{9} a^{6} + \frac{17}{9} a^{5} - \frac{20}{9} a^{4} + 2 a^{3} - \frac{16}{3} a^{2} + \frac{34}{9} a - \frac{8}{9} \),  \( \frac{1}{9} a^{7} - \frac{4}{9} a^{6} + \frac{2}{9} a^{5} - \frac{23}{9} a^{4} + 2 a^{3} - \frac{10}{3} a^{2} + \frac{40}{9} a - \frac{11}{9} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4.03157028896 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_4\wr C_2$ (as 8T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 288
The 14 conjugacy class representatives for $A_4\wr C_2$
Character table for $A_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \)

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 sibling: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: 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 R ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ R ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.8.8.12$x^{8} + 2 x^{5} + 2 x^{4} + 4$$4$$2$$8$$A_4\wr C_2$$[4/3, 4/3, 4/3, 4/3]_{3}^{6}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.5_19.6t1.1c1$1$ $ 5 \cdot 19 $ $x^{6} - x^{5} - 16 x^{4} + x^{3} + 47 x^{2} + 10 x - 11$ $C_6$ (as 6T1) $0$ $1$
1.19.3t1.1c1$1$ $ 19 $ $x^{3} - x^{2} - 6 x + 7$ $C_3$ (as 3T1) $0$ $1$
1.19.3t1.1c2$1$ $ 19 $ $x^{3} - x^{2} - 6 x + 7$ $C_3$ (as 3T1) $0$ $1$
1.5_19.6t1.1c2$1$ $ 5 \cdot 19 $ $x^{6} - x^{5} - 16 x^{4} + x^{3} + 47 x^{2} + 10 x - 11$ $C_6$ (as 6T1) $0$ $1$
2.2e2_5_19e2.3t2.1c1$2$ $ 2^{2} \cdot 5 \cdot 19^{2}$ $x^{3} - x^{2} - 25 x + 45$ $S_3$ (as 3T2) $1$ $2$
2.2e2_5_19.6t5.1c1$2$ $ 2^{2} \cdot 5 \cdot 19 $ $x^{6} - x^{5} - 6 x^{4} + 7 x^{3} + 4 x^{2} - 5 x + 1$ $S_3\times C_3$ (as 6T5) $0$ $2$
2.2e2_5_19.6t5.1c2$2$ $ 2^{2} \cdot 5 \cdot 19 $ $x^{6} - x^{5} - 6 x^{4} + 7 x^{3} + 4 x^{2} - 5 x + 1$ $S_3\times C_3$ (as 6T5) $0$ $2$
* 6.2e8_5e3_19e2.8t42.1c1$6$ $ 2^{8} \cdot 5^{3} \cdot 19^{2}$ $x^{8} + 4 x^{6} - 6 x^{5} + 7 x^{4} - 12 x^{3} + 10 x^{2} - 4 x + 1$ $A_4\wr C_2$ (as 8T42) $1$ $-2$
6.2e8_5e3_19e5.24t703.1c1$6$ $ 2^{8} \cdot 5^{3} \cdot 19^{5}$ $x^{8} + 4 x^{6} - 6 x^{5} + 7 x^{4} - 12 x^{3} + 10 x^{2} - 4 x + 1$ $A_4\wr C_2$ (as 8T42) $0$ $-2$
6.2e8_5e3_19e5.24t703.1c2$6$ $ 2^{8} \cdot 5^{3} \cdot 19^{5}$ $x^{8} + 4 x^{6} - 6 x^{5} + 7 x^{4} - 12 x^{3} + 10 x^{2} - 4 x + 1$ $A_4\wr C_2$ (as 8T42) $0$ $-2$
9.2e12_5e3_19e6.12t128.1c1$9$ $ 2^{12} \cdot 5^{3} \cdot 19^{6}$ $x^{8} + 4 x^{6} - 6 x^{5} + 7 x^{4} - 12 x^{3} + 10 x^{2} - 4 x + 1$ $A_4\wr C_2$ (as 8T42) $1$ $1$
9.2e12_5e6_19e6.18t112.1c1$9$ $ 2^{12} \cdot 5^{6} \cdot 19^{6}$ $x^{8} + 4 x^{6} - 6 x^{5} + 7 x^{4} - 12 x^{3} + 10 x^{2} - 4 x + 1$ $A_4\wr C_2$ (as 8T42) $1$ $1$

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 *.