magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![343, -327, 102, -27, 30, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 + 30*x^4 - 27*x^3 + 102*x^2 - 327*x + 343)
gp: K = bnfinit(x^6 - 3*x^5 + 30*x^4 - 27*x^3 + 102*x^2 - 327*x + 343, 1)
Normalized defining polynomial
\( x^{6} - 3 x^{5} + 30 x^{4} - 27 x^{3} + 102 x^{2} - 327 x + 343 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $6$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 3]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-720030384=-\,2^{4}\cdot 3^{8}\cdot 19^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $29.94$ | 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: | $2, 3, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{19} a^{4} - \frac{9}{19} a^{3} - \frac{1}{19} a^{2} + \frac{3}{19} a - \frac{2}{19}$, $\frac{1}{6137} a^{5} - \frac{63}{6137} a^{4} - \frac{2327}{6137} a^{3} - \frac{82}{323} a^{2} + \frac{1527}{6137} a + \frac{108}{6137}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{3}\times C_{3}$, which has order $9$
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $2$ | 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: | \( \frac{181}{6137} a^{5} - \frac{98}{6137} a^{4} + \frac{4850}{6137} a^{3} + \frac{390}{323} a^{2} + \frac{28000}{6137} a - \frac{3062}{6137} \), \( \frac{49}{6137} a^{5} - \frac{180}{6137} a^{4} + \frac{965}{6137} a^{3} + \frac{28}{323} a^{2} - \frac{14648}{6137} a + \frac{17889}{6137} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 22.1911292463 \) | 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 6 |
| The 3 conjugacy class representatives for $S_3$ |
| Character table for $S_3$ |
Intermediate fields
| \(\Q(\sqrt{-19}) \), 3.1.6156.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 3.1.6156.1 $\times$ \(\Q\) $\times$ \(\Q\) $\times$ \(\Q\) |
| Degree 3 sibling: | 3.1.6156.1 |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/7.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ | R | ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.6.4.1 | $x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| $3$ | 3.6.8.5 | $x^{6} + 9 x^{2} + 9$ | $3$ | $2$ | $8$ | $S_3$ | $[2]^{2}$ |
| $19$ | 19.2.1.2 | $x^{2} + 76$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 19.2.1.2 | $x^{2} + 76$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 76$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |