magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![746656, 21020, 16444, -211, 155, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 155*x^4 - 211*x^3 + 16444*x^2 + 21020*x + 746656)
gp: K = bnfinit(x^6 - x^5 + 155*x^4 - 211*x^3 + 16444*x^2 + 21020*x + 746656, 1)
Normalized defining polynomial
\( x^{6} - x^{5} + 155 x^{4} - 211 x^{3} + 16444 x^{2} + 21020 x + 746656 \)
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: | \(-33083784209567=-\,7^{5}\cdot 13^{4}\cdot 41^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $179.17$ | 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: | $7, 13, 41$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(3731=7\cdot 13\cdot 41\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{3731}(1,·)$, $\chi_{3731}(1108,·)$, $\chi_{3731}(165,·)$, $\chi_{3731}(614,·)$, $\chi_{3731}(1270,·)$, $\chi_{3731}(573,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{14} a^{3} + \frac{3}{14} a^{2} - \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{56} a^{4} - \frac{1}{28} a^{3} - \frac{5}{56} a^{2} + \frac{1}{4} a + \frac{2}{7}$, $\frac{1}{732872} a^{5} - \frac{193}{366436} a^{4} + \frac{17895}{732872} a^{3} + \frac{10075}{366436} a^{2} + \frac{8093}{183218} a - \frac{24096}{91609}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{3}\times C_{6342}$, which has order $19026$
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: | \( \frac{37}{366436} a^{5} - \frac{1195}{366436} a^{4} + \frac{7765}{366436} a^{3} - \frac{15015}{52348} a^{2} + \frac{297881}{183218} a - \frac{2712281}{91609} \), \( \frac{17}{183218} a^{5} - \frac{37}{366436} a^{4} + \frac{1607}{91609} a^{3} - \frac{8511}{366436} a^{2} + \frac{20661}{26174} a - \frac{630829}{91609} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 62.4891977899 \) | 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
| \(\Q(\sqrt{-287}) \), 3.3.8281.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | \(\Q\) $\times$ \(\Q(\sqrt{-287}) \) $\times$ 3.3.8281.1 |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }$ | R | ${\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }$ | ${\href{/LocalNumberField/31.6.0.1}{6} }$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ | R | ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $7$ | 7.6.5.3 | $x^{6} - 112$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| $13$ | 13.3.2.2 | $x^{3} - 13$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 13.3.2.2 | $x^{3} - 13$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| $41$ | 41.6.3.2 | $x^{6} - 1681 x^{2} + 895973$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |