Properties

Label 8.4.148132260953.1
Degree $8$
Signature $[4, 2]$
Discriminant $17^{7}\cdot 19^{2}$
Root discriminant $24.91$
Ramified primes $17, 19$
Class number $2$
Class group $[2]$
Galois Group $C_8:C_2$ (as 8T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-67, 106, 58, 24, -19, 6, -7, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 7*x^6 + 6*x^5 - 19*x^4 + 24*x^3 + 58*x^2 + 106*x - 67)
gp: K = bnfinit(x^8 - x^7 - 7*x^6 + 6*x^5 - 19*x^4 + 24*x^3 + 58*x^2 + 106*x - 67, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut x^{7} \) \(\mathstrut -\mathstrut 7 x^{6} \) \(\mathstrut +\mathstrut 6 x^{5} \) \(\mathstrut -\mathstrut 19 x^{4} \) \(\mathstrut +\mathstrut 24 x^{3} \) \(\mathstrut +\mathstrut 58 x^{2} \) \(\mathstrut +\mathstrut 106 x \) \(\mathstrut -\mathstrut 67 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[4, 2]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(148132260953=17^{7}\cdot 19^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $24.91$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $17, 19$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{512291} a^{7} - \frac{200982}{512291} a^{6} - \frac{69724}{512291} a^{5} - \frac{8764}{512291} a^{4} + \frac{141007}{512291} a^{3} + \frac{210277}{512291} a^{2} - \frac{235634}{512291} a + \frac{240147}{512291}$

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

Class group and class number

Multiplicative Abelian group isomorphic to C2, order $2$

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $5$
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:  \( \frac{1639}{512291} a^{7} - \frac{6385}{512291} a^{6} - \frac{36743}{512291} a^{5} - \frac{20048}{512291} a^{4} + \frac{67232}{512291} a^{3} + \frac{384451}{512291} a^{2} + \frac{575579}{512291} a + \frac{673736}{512291} \),  \( \frac{5776}{512291} a^{7} - \frac{20626}{512291} a^{6} - \frac{65098}{512291} a^{5} + \frac{95945}{512291} a^{4} - \frac{86258}{512291} a^{3} + \frac{430282}{512291} a^{2} + \frac{1159785}{512291} a + \frac{829626}{512291} \),  \( \frac{1932}{512291} a^{7} + \frac{19354}{512291} a^{6} + \frac{25765}{512291} a^{5} - \frac{26445}{512291} a^{4} - \frac{113288}{512291} a^{3} - \frac{503890}{512291} a^{2} - \frac{842771}{512291} a - \frac{683933}{512291} \),  \( \frac{7142}{512291} a^{7} + \frac{25938}{512291} a^{6} - \frac{21956}{512291} a^{5} - \frac{92986}{512291} a^{4} - \frac{92112}{512291} a^{3} - \frac{751169}{512291} a^{2} - \frac{534384}{512291} a + \frac{491897}{512291} \),  \( \frac{719}{39407} a^{7} - \frac{589}{39407} a^{6} - \frac{5852}{39407} a^{5} + \frac{3804}{39407} a^{4} - \frac{10178}{39407} a^{3} + \frac{23911}{39407} a^{2} + \frac{29254}{39407} a + \frac{63033}{39407} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 177.70397323 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$OD_{16}$ (as 8T7):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 16
The 10 conjugacy class representatives for $C_8:C_2$
Character table for $C_8:C_2$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R R ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$17$17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
$19$19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.19.2t1.1c1$1$ $ 19 $ $x^{2} - x + 5$ $C_2$ (as 2T1) $1$ $-1$
1.17_19.2t1.1c1$1$ $ 17 \cdot 19 $ $x^{2} - x + 81$ $C_2$ (as 2T1) $1$ $-1$
* 1.17.2t1.1c1$1$ $ 17 $ $x^{2} - x - 4$ $C_2$ (as 2T1) $1$ $1$
* 1.17.4t1.1c1$1$ $ 17 $ $x^{4} - x^{3} - 6 x^{2} + x + 1$ $C_4$ (as 4T1) $0$ $1$
1.17_19.4t1.1c1$1$ $ 17 \cdot 19 $ $x^{4} - x^{3} + 79 x^{2} + x + 1616$ $C_4$ (as 4T1) $0$ $-1$
1.17_19.4t1.1c2$1$ $ 17 \cdot 19 $ $x^{4} - x^{3} + 79 x^{2} + x + 1616$ $C_4$ (as 4T1) $0$ $-1$
* 1.17.4t1.1c2$1$ $ 17 $ $x^{4} - x^{3} - 6 x^{2} + x + 1$ $C_4$ (as 4T1) $0$ $1$
* 2.17e2_19.8t7.1c1$2$ $ 17^{2} \cdot 19 $ $x^{8} - x^{7} - 7 x^{6} + 6 x^{5} - 19 x^{4} + 24 x^{3} + 58 x^{2} + 106 x - 67$ $C_8:C_2$ (as 8T7) $0$ $0$
* 2.17e2_19.8t7.1c2$2$ $ 17^{2} \cdot 19 $ $x^{8} - x^{7} - 7 x^{6} + 6 x^{5} - 19 x^{4} + 24 x^{3} + 58 x^{2} + 106 x - 67$ $C_8:C_2$ (as 8T7) $0$ $0$

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 *.