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Properties

Label 8.0.525346636864.4
Degree $8$
Signature $[0, 4]$
Discriminant $2^{6}\cdot 7^{4}\cdot 43^{4}$
Root discriminant $29.18$
Ramified primes $2, 7, 43$
Class number $1$
Class group Trivial
Galois group $C_2^3:(C_7: C_3)$ (as 8T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![128, -280, 296, -184, 64, -12, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + 2*x^6 - 12*x^5 + 64*x^4 - 184*x^3 + 296*x^2 - 280*x + 128)
 
gp: K = bnfinit(x^8 - 2*x^7 + 2*x^6 - 12*x^5 + 64*x^4 - 184*x^3 + 296*x^2 - 280*x + 128, 1)
 

Normalized defining polynomial

\( x^{8} - 2 x^{7} + 2 x^{6} - 12 x^{5} + 64 x^{4} - 184 x^{3} + 296 x^{2} - 280 x + 128 \)

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:  \(525346636864=2^{6}\cdot 7^{4}\cdot 43^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.18$
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:  $2, 7, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{4} a^{6}$, $\frac{1}{8} a^{7}$

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:  \( \frac{1}{2} a^{3} - a^{2} + a - 1 \),  \( \frac{9}{2} a^{7} - \frac{33}{2} a^{6} - \frac{71}{2} a^{4} + 403 a^{3} - 1038 a^{2} + 1252 a - 613 \),  \( \frac{11}{4} a^{7} - \frac{27}{4} a^{6} - a^{5} - 33 a^{4} + 195 a^{3} - 469 a^{2} + 659 a - 387 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1572.57786938 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_8:C_3$ (as 8T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 168
The 8 conjugacy class representatives for $C_2^3:(C_7: C_3)$
Character table for $C_2^3:(C_7: C_3)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 14 sibling: data not computed
Degree 24 sibling: data not computed
Degree 28 sibling: data not computed
Degree 42 sibling: 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 ${\href{/LocalNumberField/3.7.0.1}{7} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ R ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.7_43.3t1.2c1$1$ $ 7 \cdot 43 $ $x^{3} - x^{2} - 100 x + 379$ $C_3$ (as 3T1) $0$ $1$
1.7_43.3t1.2c2$1$ $ 7 \cdot 43 $ $x^{3} - x^{2} - 100 x + 379$ $C_3$ (as 3T1) $0$ $1$
3.2e3_7e2_43e2.7t3.1c1$3$ $ 2^{3} \cdot 7^{2} \cdot 43^{2}$ $x^{7} - x^{6} - 25 x^{5} + 49 x^{4} + 131 x^{3} - 415 x^{2} + 281 x + 7$ $C_7:C_3$ (as 7T3) $0$ $3$
3.2e3_7e2_43e2.7t3.1c2$3$ $ 2^{3} \cdot 7^{2} \cdot 43^{2}$ $x^{7} - x^{6} - 25 x^{5} + 49 x^{4} + 131 x^{3} - 415 x^{2} + 281 x + 7$ $C_7:C_3$ (as 7T3) $0$ $3$
* 7.2e6_7e4_43e4.8t36.2c1$7$ $ 2^{6} \cdot 7^{4} \cdot 43^{4}$ $x^{8} - 2 x^{7} + 2 x^{6} - 12 x^{5} + 64 x^{4} - 184 x^{3} + 296 x^{2} - 280 x + 128$ $C_2^3:(C_7: C_3)$ (as 8T36) $1$ $-1$
7.2e6_7e5_43e5.24t283.2c1$7$ $ 2^{6} \cdot 7^{5} \cdot 43^{5}$ $x^{8} - 2 x^{7} + 2 x^{6} - 12 x^{5} + 64 x^{4} - 184 x^{3} + 296 x^{2} - 280 x + 128$ $C_2^3:(C_7: C_3)$ (as 8T36) $0$ $-1$
7.2e6_7e5_43e5.24t283.2c2$7$ $ 2^{6} \cdot 7^{5} \cdot 43^{5}$ $x^{8} - 2 x^{7} + 2 x^{6} - 12 x^{5} + 64 x^{4} - 184 x^{3} + 296 x^{2} - 280 x + 128$ $C_2^3:(C_7: C_3)$ (as 8T36) $0$ $-1$

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 *.