Properties

Label 8.4.6565418768.1
Degree $8$
Signature $[4, 2]$
Discriminant $2^{4}\cdot 17^{7}$
Root discriminant $16.87$
Ramified primes $2, 17$
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![-1, 10, -31, 35, -26, 9, 5, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 + 5*x^6 + 9*x^5 - 26*x^4 + 35*x^3 - 31*x^2 + 10*x - 1)
gp: K = bnfinit(x^8 - 3*x^7 + 5*x^6 + 9*x^5 - 26*x^4 + 35*x^3 - 31*x^2 + 10*x - 1, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut 3 x^{7} \) \(\mathstrut +\mathstrut 5 x^{6} \) \(\mathstrut +\mathstrut 9 x^{5} \) \(\mathstrut -\mathstrut 26 x^{4} \) \(\mathstrut +\mathstrut 35 x^{3} \) \(\mathstrut -\mathstrut 31 x^{2} \) \(\mathstrut +\mathstrut 10 x \) \(\mathstrut -\mathstrut 1 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[4, 2]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(6565418768=2^{4}\cdot 17^{7}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $16.87$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{13} a^{6} + \frac{2}{13} a^{5} + \frac{3}{13} a^{4} + \frac{3}{13} a^{2} - \frac{2}{13} a + \frac{1}{13}$, $\frac{1}{13} a^{7} - \frac{1}{13} a^{5} - \frac{6}{13} a^{4} + \frac{3}{13} a^{3} + \frac{5}{13} a^{2} + \frac{5}{13} a - \frac{2}{13}$

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:  \( \frac{31}{13} a^{7} - \frac{86}{13} a^{6} + \frac{135}{13} a^{5} + \frac{310}{13} a^{4} - \frac{739}{13} a^{3} + \frac{911}{13} a^{2} - \frac{752}{13} a + \frac{125}{13} \),  \( 3 a^{7} - \frac{109}{13} a^{6} + \frac{172}{13} a^{5} + \frac{388}{13} a^{4} - 72 a^{3} + \frac{1168}{13} a^{2} - \frac{965}{13} a + \frac{190}{13} \),  \( \frac{10}{13} a^{7} - \frac{30}{13} a^{6} + \frac{47}{13} a^{5} + \frac{97}{13} a^{4} - \frac{269}{13} a^{3} + \frac{311}{13} a^{2} - \frac{254}{13} a + \frac{54}{13} \),  \( \frac{31}{13} a^{7} - \frac{83}{13} a^{6} + \frac{128}{13} a^{5} + \frac{319}{13} a^{4} - \frac{700}{13} a^{3} + \frac{855}{13} a^{2} - \frac{706}{13} a + \frac{115}{13} \),  \( a \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 51.2829434823 \)
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 R ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e2.2t1.1c1$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
1.2e2_17.2t1.1c1$1$ $ 2^{2} \cdot 17 $ $x^{2} + 17$ $C_2$ (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) $0$ $1$
1.2e2_17.4t1.1c1$1$ $ 2^{2} \cdot 17 $ $x^{4} + 17 x^{2} + 68$ $C_4$ (as 4T1) $0$ $-1$
1.2e2_17.4t1.1c2$1$ $ 2^{2} \cdot 17 $ $x^{4} + 17 x^{2} + 68$ $C_4$ (as 4T1) $0$ $-1$
* 1.17.4t1.1c2$1$ $ 17 $ $x^{4} - x^{3} - 6 x^{2} + x + 1$ $C_4$ (as 4T1) $0$ $1$
* 2.2e2_17e2.8t7.1c1$2$ $ 2^{2} \cdot 17^{2}$ $x^{8} - 3 x^{7} + 5 x^{6} + 9 x^{5} - 26 x^{4} + 35 x^{3} - 31 x^{2} + 10 x - 1$ $C_8:C_2$ (as 8T7) $0$ $0$
* 2.2e2_17e2.8t7.1c2$2$ $ 2^{2} \cdot 17^{2}$ $x^{8} - 3 x^{7} + 5 x^{6} + 9 x^{5} - 26 x^{4} + 35 x^{3} - 31 x^{2} + 10 x - 1$ $C_8:C_2$ (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 *.