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Properties

Label 8.8.132705746944.1
Degree $8$
Signature $[8, 0]$
Discriminant $2^{16}\cdot 1423^{2}$
Root discriminant $24.57$
Ramified primes $2, 1423$
Class number $1$
Class group Trivial
Galois group $C_2^3:\GL(3,2)$ (as 8T48)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -8, -10, 20, 27, -4, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 10*x^6 - 4*x^5 + 27*x^4 + 20*x^3 - 10*x^2 - 8*x - 1)
 
gp: K = bnfinit(x^8 - 10*x^6 - 4*x^5 + 27*x^4 + 20*x^3 - 10*x^2 - 8*x - 1, 1)
 

Normalized defining polynomial

\( x^{8} - 10 x^{6} - 4 x^{5} + 27 x^{4} + 20 x^{3} - 10 x^{2} - 8 x - 1 \)

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:  \(132705746944=2^{16}\cdot 1423^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.57$
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:  $2, 1423$
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:  \( a \),  \( a + 1 \),  \( a^{7} - 10 a^{5} - 4 a^{4} + 28 a^{3} + 19 a^{2} - 13 a - 7 \),  \( 2 a^{7} - a^{6} - 19 a^{5} + a^{4} + 49 a^{3} + 17 a^{2} - 18 a - 3 \),  \( a^{7} - 10 a^{5} - 4 a^{4} + 27 a^{3} + 19 a^{2} - 9 a - 5 \),  \( a^{7} - 11 a^{5} - 3 a^{4} + 34 a^{3} + 15 a^{2} - 19 a - 3 \),  \( 2 a^{7} - 2 a^{6} - 17 a^{5} + 8 a^{4} + 38 a^{3} + 6 a^{2} - 11 a - 2 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1356.97140968 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3:\GL(3,2)$ (as 8T48):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1344
The 11 conjugacy class representatives for $C_2^3:\GL(3,2)$
Character table for $C_2^3:\GL(3,2)$

Intermediate fields

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

Sibling fields

Degree 8 sibling: data not computed
Degree 14 siblings: data not computed
Degree 28 siblings: data not computed
Degree 42 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 ${\href{/LocalNumberField/3.7.0.1}{7} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.8.16.20$x^{8} + 2 x^{4} + 8 x^{2} + 8 x + 4$$4$$2$$16$$V_4^2:S_3$$[4/3, 4/3, 8/3, 8/3]_{3}^{2}$
1423Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
3.2e8_1423e2.42t37.2c1$3$ $ 2^{8} \cdot 1423^{2}$ $x^{7} - 12 x^{5} + 34 x^{3} - 8 x^{2} - 24 x + 8$ $\GL(3,2)$ (as 7T5) $0$ $3$
3.2e8_1423e2.42t37.2c2$3$ $ 2^{8} \cdot 1423^{2}$ $x^{7} - 12 x^{5} + 34 x^{3} - 8 x^{2} - 24 x + 8$ $\GL(3,2)$ (as 7T5) $0$ $3$
6.2e10_1423e2.7t5.2c1$6$ $ 2^{10} \cdot 1423^{2}$ $x^{7} - 12 x^{5} + 34 x^{3} - 8 x^{2} - 24 x + 8$ $\GL(3,2)$ (as 7T5) $1$ $6$
* 7.2e16_1423e2.8t48.1c1$7$ $ 2^{16} \cdot 1423^{2}$ $x^{8} - 10 x^{6} - 4 x^{5} + 27 x^{4} + 20 x^{3} - 10 x^{2} - 8 x - 1$ $C_2^3:\GL(3,2)$ (as 8T48) $1$ $7$
7.2e16_1423e4.8t37.2c1$7$ $ 2^{16} \cdot 1423^{4}$ $x^{7} - 12 x^{5} + 34 x^{3} - 8 x^{2} - 24 x + 8$ $\GL(3,2)$ (as 7T5) $1$ $7$
7.2e12_1423e4.8t48.1c1$7$ $ 2^{12} \cdot 1423^{4}$ $x^{8} - 10 x^{6} - 4 x^{5} + 27 x^{4} + 20 x^{3} - 10 x^{2} - 8 x - 1$ $C_2^3:\GL(3,2)$ (as 8T48) $1$ $7$
8.2e18_1423e4.21t14.1c1$8$ $ 2^{18} \cdot 1423^{4}$ $x^{7} - x^{6} - 12 x^{5} + 18 x^{4} + 14 x^{3} - 30 x^{2} + 14 x - 2$ $\GL(3,2)$ (as 7T5) $1$ $8$
14.2e30_1423e6.28t159.1c1$14$ $ 2^{30} \cdot 1423^{6}$ $x^{8} - 10 x^{6} - 4 x^{5} + 27 x^{4} + 20 x^{3} - 10 x^{2} - 8 x - 1$ $C_2^3:\GL(3,2)$ (as 8T48) $1$ $14$
21.2e48_1423e12.42t210.1c1$21$ $ 2^{48} \cdot 1423^{12}$ $x^{8} - 10 x^{6} - 4 x^{5} + 27 x^{4} + 20 x^{3} - 10 x^{2} - 8 x - 1$ $C_2^3:\GL(3,2)$ (as 8T48) $1$ $21$
21.2e52_1423e10.42t210.1c1$21$ $ 2^{52} \cdot 1423^{10}$ $x^{8} - 10 x^{6} - 4 x^{5} + 27 x^{4} + 20 x^{3} - 10 x^{2} - 8 x - 1$ $C_2^3:\GL(3,2)$ (as 8T48) $1$ $21$

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 *.