Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + x^6 - x^4 + x^2 - x + 1)
gp: K = bnfinit(x^8 - x^7 + x^6 - x^4 + x^2 - x + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1, 0, -1, 0, 1, -1, 1]);
\(x^{8} - x^{7} + x^{6} - x^{4} + x^{2} - x + 1\)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
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Discriminant: | \(2067245\)\(\medspace = 5\cdot 643^{2}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
Root discriminant: | $6.16$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
Ramified primes: | $5, 643$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
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$|\Aut(K/\Q)|$: | $2$ | ||
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
Class group and class number
Trivial group, which has order $1$
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
Unit group
sage: UK = K.unit_group()
magma: UK, f := UnitGroup(K);
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
Torsion generator: | \( -1 \) (order $2$) ![]() | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
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Fundamental units: | \( a^{7} - a^{6} + a^{5} - a^{3} + a - 1 \), \( a^{6} - a^{5} - a^{2} + 1 \), \( a^{4} + a - 1 \) ![]() | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
Regulator: | \( 0.453534943932 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$C_2^3:S_4.C_2$ (as 8T44):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
A solvable group of order 384 |
The 20 conjugacy class representatives for $C_2 \wr S_4$ |
Character table for $C_2 \wr S_4$ |
Intermediate fields
4.2.643.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 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 | ${\href{/LocalNumberField/2.8.0.1}{8} }$ | ${\href{/LocalNumberField/3.8.0.1}{8} }$ | R | ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }$ | ${\href{/LocalNumberField/17.8.0.1}{8} }$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
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]])
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];
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
$5$ | 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
5.4.0.1 | $x^{4} + x^{2} - 2 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
643 | Data not computed |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.643.2t1.a.a | $1$ | $ 643 $ | \(\Q(\sqrt{-643}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.3215.2t1.a.a | $1$ | $ 5 \cdot 643 $ | \(\Q(\sqrt{-3215}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.643.3t2.a.a | $2$ | $ 643 $ | 3.1.643.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.16075.6t3.b.a | $2$ | $ 5^{2} \cdot 643 $ | 6.0.33230963375.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 3.643.4t5.a.a | $3$ | $ 643 $ | 4.2.643.1 | $S_4$ (as 4T5) | $1$ | $1$ |
3.413449.6t8.a.a | $3$ | $ 643^{2}$ | 4.2.643.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.80375.6t11.a.a | $3$ | $ 5^{3} \cdot 643 $ | 6.2.51681125.1 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
3.51681125.6t11.a.a | $3$ | $ 5^{3} \cdot 643^{2}$ | 6.2.51681125.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
* | 4.3215.8t44.b.a | $4$ | $ 5 \cdot 643 $ | 8.0.2067245.1 | $C_2 \wr S_4$ (as 8T44) | $1$ | $-2$ |
4.80375.8t44.b.a | $4$ | $ 5^{3} \cdot 643 $ | 8.0.2067245.1 | $C_2 \wr S_4$ (as 8T44) | $1$ | $-2$ | |
4.1329238535.8t44.b.a | $4$ | $ 5 \cdot 643^{3}$ | 8.0.2067245.1 | $C_2 \wr S_4$ (as 8T44) | $1$ | $2$ | |
4.33230963375.8t44.b.a | $4$ | $ 5^{3} \cdot 643^{3}$ | 8.0.2067245.1 | $C_2 \wr S_4$ (as 8T44) | $1$ | $2$ | |
6.51681125.8t41.b.a | $6$ | $ 5^{3} \cdot 643^{2}$ | 8.4.258405625.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $2$ | |
6.213...125.12t108.b.a | $6$ | $ 5^{3} \cdot 643^{4}$ | 8.4.258405625.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $-2$ | |
6.33230963375.8t41.b.a | $6$ | $ 5^{3} \cdot 643^{3}$ | 8.4.258405625.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $0$ | |
6.33230963375.12t108.b.a | $6$ | $ 5^{3} \cdot 643^{3}$ | 8.4.258405625.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $0$ | |
8.267...625.24t708.b.a | $8$ | $ 5^{6} \cdot 643^{4}$ | 8.0.2067245.1 | $C_2 \wr S_4$ (as 8T44) | $1$ | $0$ | |
8.427...025.24t708.b.a | $8$ | $ 5^{2} \cdot 643^{4}$ | 8.0.2067245.1 | $C_2 \wr S_4$ (as 8T44) | $1$ | $0$ |
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 *.