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Properties

Label 8.0.151803769808.4
Degree $8$
Signature $[0, 4]$
Discriminant $2^{4}\cdot 29^{3}\cdot 73^{3}$
Root discriminant $24.98$
Ramified primes $2, 29, 73$
Class number $6$
Class group $[6]$
Galois group $\textrm{GL(2,3)}$ (as 8T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -64, 80, -28, 30, -7, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 5*x^6 - 7*x^5 + 30*x^4 - 28*x^3 + 80*x^2 - 64*x + 256)
 
gp: K = bnfinit(x^8 - x^7 + 5*x^6 - 7*x^5 + 30*x^4 - 28*x^3 + 80*x^2 - 64*x + 256, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 5 x^{6} - 7 x^{5} + 30 x^{4} - 28 x^{3} + 80 x^{2} - 64 x + 256 \)

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:  \(151803769808=2^{4}\cdot 29^{3}\cdot 73^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.98$
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:  $2, 29, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{32} a^{6} - \frac{1}{32} a^{5} - \frac{3}{32} a^{4} - \frac{7}{32} a^{3} + \frac{3}{16} a^{2} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{256} a^{7} + \frac{3}{256} a^{6} + \frac{17}{256} a^{5} - \frac{3}{256} a^{4} + \frac{9}{128} a^{3} - \frac{5}{64} a^{2} - \frac{1}{2} a - \frac{1}{4}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{6}$, which has order $6$

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:  \( \frac{1}{64} a^{7} + \frac{1}{64} a^{6} + \frac{3}{64} a^{5} + \frac{3}{64} a^{4} + \frac{1}{4} a^{3} - \frac{1}{8} a - \frac{1}{2} \),  \( \frac{1}{256} a^{7} - \frac{5}{256} a^{6} + \frac{25}{256} a^{5} - \frac{43}{256} a^{4} + \frac{37}{128} a^{3} - \frac{33}{64} a^{2} + \frac{7}{8} a - \frac{7}{4} \),  \( \frac{5}{256} a^{7} - \frac{1}{256} a^{6} + \frac{37}{256} a^{5} - \frac{31}{256} a^{4} + \frac{69}{128} a^{3} - \frac{33}{64} a^{2} + \frac{3}{4} a - \frac{5}{4} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 75.1200451982 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\GL(2,3)$ (as 8T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 8 conjugacy class representatives for $\textrm{GL(2,3)}$
Character table for $\textrm{GL(2,3)}$

Intermediate fields

4.4.8468.1

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

Sibling fields

Degree 16 sibling: data not computed
Degree 24 sibling: data not computed
Arithmetically equvalently sibling: 8.0.151803769808.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{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}$
73Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.29_73.2t1.1c1$1$ $ 29 \cdot 73 $ $x^{2} - x - 529$ $C_2$ (as 2T1) $1$ $1$
2.2e2_29_73.3t2.1c1$2$ $ 2^{2} \cdot 29 \cdot 73 $ $x^{3} - x^{2} - 27 x - 43$ $S_3$ (as 3T2) $1$ $2$
2.2e2_29_73.24t22.4c1$2$ $ 2^{2} \cdot 29 \cdot 73 $ $x^{8} - x^{7} + 5 x^{6} - 7 x^{5} + 30 x^{4} - 28 x^{3} + 80 x^{2} - 64 x + 256$ $\textrm{GL(2,3)}$ (as 8T23) $0$ $-2$
2.2e2_29_73.24t22.4c2$2$ $ 2^{2} \cdot 29 \cdot 73 $ $x^{8} - x^{7} + 5 x^{6} - 7 x^{5} + 30 x^{4} - 28 x^{3} + 80 x^{2} - 64 x + 256$ $\textrm{GL(2,3)}$ (as 8T23) $0$ $-2$
3.2e2_29e2_73e2.6t8.1c1$3$ $ 2^{2} \cdot 29^{2} \cdot 73^{2}$ $x^{4} - x^{3} - 5 x^{2} + 3 x + 4$ $S_4$ (as 4T5) $1$ $3$
* 3.2e2_29_73.4t5.1c1$3$ $ 2^{2} \cdot 29 \cdot 73 $ $x^{4} - x^{3} - 5 x^{2} + 3 x + 4$ $S_4$ (as 4T5) $1$ $3$
* 4.2e2_29e2_73e2.8t23.4c1$4$ $ 2^{2} \cdot 29^{2} \cdot 73^{2}$ $x^{8} - x^{7} + 5 x^{6} - 7 x^{5} + 30 x^{4} - 28 x^{3} + 80 x^{2} - 64 x + 256$ $\textrm{GL(2,3)}$ (as 8T23) $1$ $-4$

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 *.