magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1161, -550, 389, 98, -44, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - 44*x^4 + 98*x^3 + 389*x^2 - 550*x - 1161)
gp: K = bnfinit(x^6 - 2*x^5 - 44*x^4 + 98*x^3 + 389*x^2 - 550*x - 1161, 1)
Normalized defining polynomial
\( x^{6} - 2 x^{5} - 44 x^{4} + 98 x^{3} + 389 x^{2} - 550 x - 1161 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $6$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[6, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(7485795513=3^{2}\cdot 7^{2}\cdot 257^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $44.23$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $3, 7, 257$ | magma: PrimeDivisors(Discriminant(Integers(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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{6} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{4} - \frac{1}{2} a^{2} - \frac{1}{6} a$, $\frac{1}{342} a^{5} - \frac{11}{171} a^{4} - \frac{1}{114} a^{3} - \frac{13}{342} a^{2} + \frac{68}{171} a + \frac{2}{19}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{4}$, which has order $4$
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: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 586.866867247 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 24 |
| The 5 conjugacy class representatives for $S_4$ |
| Character table for $S_4$ |
Intermediate fields
| 3.3.257.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Galois closure: | data not computed |
| Twin sextic algebra: | data not computed |
| Degree 4 sibling: | 4.4.113337.1 |
| Degree 6 sibling: | 6.6.29127609.1 |
| Degree 8 sibling: | 8.8.848417606056881.2 |
| Degree 12 siblings: | Deg 12, Deg 12 |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ | R | ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $3$ | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 3.4.2.2 | $x^{4} - 3 x^{2} + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| $7$ | 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 257 | Data not computed | ||||||