magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8977968, 2342628, 167112, -4317, -765, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 - 765*x^4 - 4317*x^3 + 167112*x^2 + 2342628*x + 8977968)
gp: K = bnfinit(x^6 - 3*x^5 - 765*x^4 - 4317*x^3 + 167112*x^2 + 2342628*x + 8977968, 1)
Normalized defining polynomial
\( x^{6} - 3 x^{5} - 765 x^{4} - 4317 x^{3} + 167112 x^{2} + 2342628 x + 8977968 \)
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: | \(-13530938286500271=-\,3^{9}\cdot 7^{4}\cdot 13^{3}\cdot 19^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $488.15$ | 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: | $3, 7, 13, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(15561=3^{2}\cdot 7\cdot 13\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{15561}(1,·)$, $\chi_{15561}(3355,·)$, $\chi_{15561}(4757,·)$, $\chi_{15561}(5422,·)$, $\chi_{15561}(7877,·)$, $\chi_{15561}(9710,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{4} - \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{6875424} a^{5} - \frac{10903}{1145904} a^{4} - \frac{20261}{2291808} a^{3} + \frac{81221}{572952} a^{2} + \frac{23115}{190984} a + \frac{4419}{47746}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{4}\times C_{12}\times C_{4332}$, which has order $207936$ (assuming GRH)
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: | \( \frac{73}{1145904} a^{5} - \frac{457}{572952} a^{4} - \frac{44527}{1145904} a^{3} + \frac{8629}{95492} a^{2} + \frac{669553}{95492} a + \frac{752904}{23873} \), \( \frac{201}{381968} a^{5} - \frac{4429}{572952} a^{4} - \frac{365189}{1145904} a^{3} + \frac{77989}{47746} a^{2} + \frac{6721725}{95492} a + \frac{9068256}{23873} \) (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 127.01361460330098 \) (assuming GRH) | 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 cyclic group of order 6 |
| The 6 conjugacy class representatives for $C_6$ |
| Character table for $C_6$ |
Intermediate fields
| \(\Q(\sqrt{-39}) \), 3.3.1432809.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 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 | ${\href{/LocalNumberField/2.1.0.1}{1} }^{6}$ | R | ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ | R | ${\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ | R | ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ | R | ${\href{/LocalNumberField/23.6.0.1}{6} }$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }$ | ${\href{/LocalNumberField/37.6.0.1}{6} }$ | ${\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.6.9.10 | $x^{6} + 6 x^{4} + 3$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ |
| $7$ | 7.6.4.2 | $x^{6} - 7 x^{3} + 147$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| $13$ | 13.6.3.1 | $x^{6} - 52 x^{4} + 676 x^{2} - 79092$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| $19$ | 19.6.4.1 | $x^{6} + 57 x^{3} + 1444$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |