Properties

Label 8.4.186638688289.1
Degree $8$
Signature $[4, 2]$
Discriminant $41^{4}\cdot 257^{2}$
Root discriminant $25.64$
Ramified primes $41, 257$
Class number $1$
Class group Trivial
Galois Group $V_4^2:(S_3\times C_2)$ (as 8T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-10, 17, 1, -13, 17, 6, -8, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 8*x^6 + 6*x^5 + 17*x^4 - 13*x^3 + x^2 + 17*x - 10)
gp: K = bnfinit(x^8 - x^7 - 8*x^6 + 6*x^5 + 17*x^4 - 13*x^3 + x^2 + 17*x - 10, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut x^{7} \) \(\mathstrut -\mathstrut 8 x^{6} \) \(\mathstrut +\mathstrut 6 x^{5} \) \(\mathstrut +\mathstrut 17 x^{4} \) \(\mathstrut -\mathstrut 13 x^{3} \) \(\mathstrut +\mathstrut x^{2} \) \(\mathstrut +\mathstrut 17 x \) \(\mathstrut -\mathstrut 10 \)

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:  \(186638688289=41^{4}\cdot 257^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $25.64$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $41, 257$
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 \(a\))

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

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:  $5$
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:  \( a^{2} + a - 1 \),  \( a^{2} - a - 3 \),  \( \frac{1}{2} a^{7} - \frac{3}{2} a^{6} - \frac{3}{2} a^{5} + 6 a^{4} - \frac{3}{2} a^{3} - 3 a^{2} + 6 a - 3 \),  \( a^{7} - 5 a^{5} - a^{4} + 2 a^{3} - 5 a^{2} + 1 \),  \( a^{7} - \frac{5}{2} a^{6} - \frac{9}{2} a^{5} + \frac{27}{2} a^{4} - 2 a^{3} - \frac{29}{2} a^{2} + 22 a - 9 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 1965.77880514 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2^2:S_4:C_2$ (as 8T41):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 192
The 14 conjugacy class representatives for $V_4^2:(S_3\times C_2)$
Character table for $V_4^2:(S_3\times C_2)$

Intermediate fields

\(\Q(\sqrt{41}) \)

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

Sibling fields

Degree 8 sibling: data not computed
Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$41$41.8.4.1$x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
257Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.41.2t1.1c1$1$ $ 41 $ $x^{2} - x - 10$ $C_2$ (as 2T1) $1$ $1$
1.41_257.2t1.1c1$1$ $ 41 \cdot 257 $ $x^{2} - x - 2634$ $C_2$ (as 2T1) $1$ $1$
1.257.2t1.1c1$1$ $ 257 $ $x^{2} - x - 64$ $C_2$ (as 2T1) $1$ $1$
2.41e2_257.6t3.1c1$2$ $ 41^{2} \cdot 257 $ $x^{6} - x^{5} - 39 x^{4} + 23 x^{3} + 329 x^{2} - 195 x - 333$ $D_{6}$ (as 6T3) $1$ $2$
2.257.3t2.1c1$2$ $ 257 $ $x^{3} - x^{2} - 4 x + 3$ $S_3$ (as 3T2) $1$ $2$
3.257e2.6t8.1c1$3$ $ 257^{2}$ $x^{4} + x^{2} - x + 1$ $S_4$ (as 4T5) $1$ $-1$
3.41e3_257.6t11.1c1$3$ $ 41^{3} \cdot 257 $ $x^{6} - x^{5} + 27 x^{4} - 69 x^{3} - 5885 x^{2} + 5677 x - 298255$ $S_4\times C_2$ (as 6T11) $1$ $-1$
3.41e3_257e2.6t11.1c1$3$ $ 41^{3} \cdot 257^{2}$ $x^{6} - x^{5} + 27 x^{4} - 69 x^{3} - 5885 x^{2} + 5677 x - 298255$ $S_4\times C_2$ (as 6T11) $1$ $-1$
3.257.4t5.1c1$3$ $ 257 $ $x^{4} + x^{2} - x + 1$ $S_4$ (as 4T5) $1$ $-1$
6.41e3_257e3.12t108.1c1$6$ $ 41^{3} \cdot 257^{3}$ $x^{8} - x^{7} - 8 x^{6} + 6 x^{5} + 17 x^{4} - 13 x^{3} + x^{2} + 17 x - 10$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $-2$
* 6.41e3_257e2.8t41.1c1$6$ $ 41^{3} \cdot 257^{2}$ $x^{8} - x^{7} - 8 x^{6} + 6 x^{5} + 17 x^{4} - 13 x^{3} + x^{2} + 17 x - 10$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $2$
6.41e3_257e4.12t111.1c1$6$ $ 41^{3} \cdot 257^{4}$ $x^{8} - x^{7} - 8 x^{6} + 6 x^{5} + 17 x^{4} - 13 x^{3} + x^{2} + 17 x - 10$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $2$
6.41e3_257e3.8t41.1c1$6$ $ 41^{3} \cdot 257^{3}$ $x^{8} - x^{7} - 8 x^{6} + 6 x^{5} + 17 x^{4} - 13 x^{3} + x^{2} + 17 x - 10$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $-2$

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 *.