magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19471804416, 1446882480, 27314802, -262833, -10359, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 - 10359*x^4 - 262833*x^3 + 27314802*x^2 + 1446882480*x + 19471804416)
gp: K = bnfinit(x^6 - 3*x^5 - 10359*x^4 - 262833*x^3 + 27314802*x^2 + 1446882480*x + 19471804416, 1)
Normalized defining polynomial
\( x^{6} - 3 x^{5} - 10359 x^{4} - 262833 x^{3} + 27314802 x^{2} + 1446882480 x + 19471804416 \)
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: | \(-21987774715562940375=-\,3^{9}\cdot 5^{3}\cdot 7^{4}\cdot 13^{4}\cdot 19^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $1673.77$ | 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: | $3, 5, 7, 13, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(77805=3^{2}\cdot 5\cdot 7\cdot 13\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{77805}(1,·)$, $\chi_{77805}(2804,·)$, $\chi_{77805}(4111,·)$, $\chi_{77805}(12104,·)$, $\chi_{77805}(16636,·)$, $\chi_{77805}(42149,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{48} a^{4} + \frac{3}{16} a^{2} - \frac{1}{8} a$, $\frac{1}{1594703232} a^{5} - \frac{805777}{132891936} a^{4} - \frac{8322433}{531567744} a^{3} + \frac{25758007}{265783872} a^{2} - \frac{544715}{1384291} a + \frac{122939}{1384291}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{54666}$, which has order $4427946$ (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: | \( \frac{6869}{132891936} a^{5} - \frac{38047}{16611492} a^{4} - \frac{58705063}{132891936} a^{3} + \frac{105186325}{22148656} a^{2} + \frac{6752798049}{5537164} a + \frac{33416006221}{1384291} \), \( \frac{12871}{265783872} a^{5} - \frac{47595}{22148656} a^{4} - \frac{109998805}{265783872} a^{3} + \frac{198488665}{44297312} a^{2} + \frac{3167990735}{2768582} a + \frac{31029502301}{1384291} \) (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 212.1071224368399 \) (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
| \(\Q(\sqrt{-15}) \), 3.3.242144721.4 |
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 | R | R | ${\href{/LocalNumberField/11.6.0.1}{6} }$ | R | ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ | R | ${\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.6.9.3 | $x^{6} + 3 x^{4} + 24$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ |
| $5$ | 5.6.3.2 | $x^{6} - 25 x^{2} + 250$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| $7$ | 7.6.4.1 | $x^{6} + 35 x^{3} + 441$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| $13$ | 13.6.4.3 | $x^{6} + 65 x^{3} + 1352$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| $19$ | 19.3.2.3 | $x^{3} - 304$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 19.3.2.3 | $x^{3} - 304$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |