Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 2*x^4 - 2*x^3 + 3*x^2 - x + 1)
gp: K = bnfinit(x^6 - x^5 + 2*x^4 - 2*x^3 + 3*x^2 - x + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 3, -2, 2, -1, 1]);
\(x^{6} - x^{5} + 2 x^{4} - 2 x^{3} + 3 x^{2} - x + 1\)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
Signature: | $[0, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
Discriminant: | \(-25747\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
Root discriminant: | $5.43$ | 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: | $25747$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
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$|\Aut(K/\Q)|$: | $1$ | ||
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$
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: | $2$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
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Torsion generator: | \( -1 \) (order $2$) ![]() | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
Fundamental units: | \( a \), \( a^{5} + a^{3} + a + 1 \) ![]() | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
Regulator: | \( 0.457055178835 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$S_6$ (as 6T16):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
A non-solvable group of order 720 |
The 11 conjugacy class representatives for $S_6$ |
Character table for $S_6$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling algebras
Twin sextic algebra: | 6.4.17067892507723.1 |
Degree 6 sibling: | 6.4.17067892507723.1 |
Degree 10 sibling: | 10.4.17067892507723.1 |
Degree 12 siblings: | Deg 12, Deg 12 |
Degree 15 siblings: | Deg 15, Deg 15 |
Degree 20 siblings: | Deg 20, Deg 20, Deg 20 |
Degree 30 siblings: | data not computed |
Degree 36 sibling: | data not computed |
Degree 40 siblings: | data not computed |
Degree 45 sibling: | 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 | ${\href{/LocalNumberField/2.6.0.1}{6} }$ | ${\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ | ${\href{/LocalNumberField/5.6.0.1}{6} }$ | ${\href{/LocalNumberField/7.6.0.1}{6} }$ | ${\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | ${\href{/LocalNumberField/13.6.0.1}{6} }$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ | ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | ${\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
25747 | Data not computed |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.25747.2t1.a.a | $1$ | $ 25747 $ | \(\Q(\sqrt{-25747}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 5.25747.6t16.a.a | $5$ | $ 25747 $ | 6.0.25747.1 | $S_6$ (as 6T16) | $1$ | $-1$ |
5.662908009.12t183.a.a | $5$ | $ 25747^{2}$ | 6.0.25747.1 | $S_6$ (as 6T16) | $1$ | $-3$ | |
5.439...081.12t183.a.a | $5$ | $ 25747^{4}$ | 6.0.25747.1 | $S_6$ (as 6T16) | $1$ | $1$ | |
5.170...723.6t16.a.a | $5$ | $ 25747^{3}$ | 6.0.25747.1 | $S_6$ (as 6T16) | $1$ | $3$ | |
9.170...723.10t32.a.a | $9$ | $ 25747^{3}$ | 6.0.25747.1 | $S_6$ (as 6T16) | $1$ | $3$ | |
9.291...729.20t145.a.a | $9$ | $ 25747^{6}$ | 6.0.25747.1 | $S_6$ (as 6T16) | $1$ | $-3$ | |
10.291...729.30t164.a.a | $10$ | $ 25747^{6}$ | 6.0.25747.1 | $S_6$ (as 6T16) | $1$ | $2$ | |
10.439...081.30t164.a.a | $10$ | $ 25747^{4}$ | 6.0.25747.1 | $S_6$ (as 6T16) | $1$ | $-2$ | |
16.193...561.36t1252.a.a | $16$ | $ 25747^{8}$ | 6.0.25747.1 | $S_6$ (as 6T16) | $1$ | $0$ |
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 *.