magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![225366016, 4609088, 575252, -3343, 251, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 251*x^4 - 3343*x^3 + 575252*x^2 + 4609088*x + 225366016)
gp: K = bnfinit(x^6 - x^5 + 251*x^4 - 3343*x^3 + 575252*x^2 + 4609088*x + 225366016, 1)
Normalized defining polynomial
\( x^{6} - x^{5} + 251 x^{4} - 3343 x^{3} + 575252 x^{2} + 4609088 x + 225366016 \)
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: | \(-827762709645597375=-\,3^{3}\cdot 5^{3}\cdot 13^{4}\cdot 97^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $968.99$ | 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: | $3, 5, 13, 97$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(18915=3\cdot 5\cdot 13\cdot 97\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{18915}(1,·)$, $\chi_{18915}(1394,·)$, $\chi_{18915}(8791,·)$, $\chi_{18915}(13906,·)$, $\chi_{18915}(16004,·)$, $\chi_{18915}(16649,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{388} a^{3} + \frac{12}{97} a^{2} - \frac{105}{388} a - \frac{43}{97}$, $\frac{1}{2328} a^{4} + \frac{307}{2328} a^{2} - \frac{47}{194} a + \frac{62}{291}$, $\frac{1}{84757824} a^{5} - \frac{2513}{84757824} a^{4} - \frac{94901}{84757824} a^{3} - \frac{18506879}{84757824} a^{2} - \frac{4066895}{21189456} a + \frac{91987}{1324341}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{2}\times C_{14}\times C_{84}\times C_{588}$, which has order $1382976$ (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: | \( \frac{105}{4708768} a^{5} - \frac{9009}{4708768} a^{4} + \frac{59731}{4708768} a^{3} + \frac{564417}{4708768} a^{2} + \frac{15533469}{1177192} a - \frac{154071279}{147149} \), \( \frac{2051}{42378912} a^{5} - \frac{57043}{42378912} a^{4} + \frac{1087457}{42378912} a^{3} - \frac{2414461}{42378912} a^{2} + \frac{115685003}{10594728} a + \frac{218760181}{1324341} \) (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 361.6273031734536 \) (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
| \(\Q(\sqrt{-1455}) \), 3.3.1590121.2 |
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 | R | ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }$ | R | ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }$ | ${\href{/LocalNumberField/23.6.0.1}{6} }$ | ${\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $5$ | 5.6.3.1 | $x^{6} - 10 x^{4} + 25 x^{2} - 500$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| $13$ | 13.3.2.3 | $x^{3} - 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 13.3.2.3 | $x^{3} - 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| $97$ | 97.6.5.4 | $x^{6} + 485$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |