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Properties

Label 8.8.309593125.1
Degree $8$
Signature $[8, 0]$
Discriminant $5^{4}\cdot 19\cdot 29^{2}\cdot 31$
Root discriminant $11.52$
Ramified primes $5, 19, 29, 31$
Class number $1$
Class group Trivial
Galois group $C_2 \wr C_2\wr C_2$ (as 8T35)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 9, 6, -23, -5, 17, -1, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 - x^6 + 17*x^5 - 5*x^4 - 23*x^3 + 6*x^2 + 9*x - 1)
 
gp: K = bnfinit(x^8 - 4*x^7 - x^6 + 17*x^5 - 5*x^4 - 23*x^3 + 6*x^2 + 9*x - 1, 1)
 

Normalized defining polynomial

\( x^{8} - 4 x^{7} - x^{6} + 17 x^{5} - 5 x^{4} - 23 x^{3} + 6 x^{2} + 9 x - 1 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(309593125=5^{4}\cdot 19\cdot 29^{2}\cdot 31\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.52$
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:  $5, 19, 29, 31$
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:  $7$
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:  \( a^{6} - 3 a^{5} - 2 a^{4} + 9 a^{3} + a^{2} - 6 a - 1 \),  \( a^{2} - a - 2 \),  \( a^{5} - 3 a^{4} - a^{3} + 7 a^{2} - a - 3 \),  \( a^{3} - 2 a^{2} - 2 a + 3 \),  \( a^{6} - 3 a^{5} - 2 a^{4} + 10 a^{3} - a^{2} - 8 a + 2 \),  \( a^{7} - 3 a^{6} - 2 a^{5} + 9 a^{4} + a^{3} - 7 a^{2} + 1 \),  \( a^{5} - 2 a^{4} - 3 a^{3} + 5 a^{2} + 2 a - 2 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 23.6967890277 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4^2.C_2$ (as 8T35):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 20 conjugacy class representatives for $C_2 \wr C_2\wr C_2$
Character table for $C_2 \wr C_2\wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1

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 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 ${\href{/LocalNumberField/2.8.0.1}{8} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ R ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.5_29.2t1.1c1$1$ $ 5 \cdot 29 $ $x^{2} - x - 36$ $C_2$ (as 2T1) $1$ $1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.29.2t1.1c1$1$ $ 29 $ $x^{2} - x - 7$ $C_2$ (as 2T1) $1$ $1$
1.5_19_29_31.2t1.1c1$1$ $ 5 \cdot 19 \cdot 29 \cdot 31 $ $x^{2} - x - 21351$ $C_2$ (as 2T1) $1$ $1$
1.19_31.2t1.1c1$1$ $ 19 \cdot 31 $ $x^{2} - x - 147$ $C_2$ (as 2T1) $1$ $1$
1.5_19_31.2t1.1c1$1$ $ 5 \cdot 19 \cdot 31 $ $x^{2} - x - 736$ $C_2$ (as 2T1) $1$ $1$
1.19_29_31.2t1.1c1$1$ $ 19 \cdot 29 \cdot 31 $ $x^{2} - x - 4270$ $C_2$ (as 2T1) $1$ $1$
2.5_19_29_31.4t3.2c1$2$ $ 5 \cdot 19 \cdot 29 \cdot 31 $ $x^{4} - 2 x^{3} - 142 x^{2} + 143 x + 842$ $D_{4}$ (as 4T3) $1$ $2$
2.5_19_29_31.4t3.1c1$2$ $ 5 \cdot 19 \cdot 29 \cdot 31 $ $x^{4} - 2 x^{3} - 130 x^{2} + 131 x + 20$ $D_{4}$ (as 4T3) $1$ $2$
2.5_19_31.4t3.2c1$2$ $ 5 \cdot 19 \cdot 31 $ $x^{4} - 2 x^{3} - 24 x^{2} + 25 x + 9$ $D_{4}$ (as 4T3) $1$ $2$
2.5_19_29e2_31.4t3.1c1$2$ $ 5 \cdot 19 \cdot 29^{2} \cdot 31 $ $x^{4} - x^{3} - 414 x^{2} - 1928 x + 10607$ $D_{4}$ (as 4T3) $1$ $2$
2.5_19e2_29_31e2.4t3.1c1$2$ $ 5 \cdot 19^{2} \cdot 29 \cdot 31^{2}$ $x^{4} - x^{3} - 1034 x^{2} - 1618 x + 107017$ $D_{4}$ (as 4T3) $1$ $2$
* 2.5_29.4t3.1c1$2$ $ 5 \cdot 29 $ $x^{4} - x^{3} - 5 x^{2} - x + 1$ $D_{4}$ (as 4T3) $1$ $2$
4.5e2_19e3_29_31e3.8t35.1c1$4$ $ 5^{2} \cdot 19^{3} \cdot 29 \cdot 31^{3}$ $x^{8} - 4 x^{7} - x^{6} + 17 x^{5} - 5 x^{4} - 23 x^{3} + 6 x^{2} + 9 x - 1$ $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $4$
4.5e3_19e2_29e2_31e2.8t29.1c1$4$ $ 5^{3} \cdot 19^{2} \cdot 29^{2} \cdot 31^{2}$ $x^{8} - x^{7} - 41 x^{6} + 2 x^{5} + 457 x^{4} + 240 x^{3} - 1302 x^{2} - 544 x + 863$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $1$ $4$
4.5e2_19_29e3_31.8t35.1c1$4$ $ 5^{2} \cdot 19 \cdot 29^{3} \cdot 31 $ $x^{8} - 4 x^{7} - x^{6} + 17 x^{5} - 5 x^{4} - 23 x^{3} + 6 x^{2} + 9 x - 1$ $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $4$
4.5_19e2_29e2_31e2.8t29.1c1$4$ $ 5 \cdot 19^{2} \cdot 29^{2} \cdot 31^{2}$ $x^{8} - x^{7} - 41 x^{6} + 2 x^{5} + 457 x^{4} + 240 x^{3} - 1302 x^{2} - 544 x + 863$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $1$ $4$
4.5e2_19e3_29e3_31e3.8t35.1c1$4$ $ 5^{2} \cdot 19^{3} \cdot 29^{3} \cdot 31^{3}$ $x^{8} - 4 x^{7} - x^{6} + 17 x^{5} - 5 x^{4} - 23 x^{3} + 6 x^{2} + 9 x - 1$ $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $4$
* 4.5e2_19_29_31.8t35.1c1$4$ $ 5^{2} \cdot 19 \cdot 29 \cdot 31 $ $x^{8} - 4 x^{7} - x^{6} + 17 x^{5} - 5 x^{4} - 23 x^{3} + 6 x^{2} + 9 x - 1$ $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $4$

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 *.