magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![184542631, -321489, 967309, -1133, 1699, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 1699*x^4 - 1133*x^3 + 967309*x^2 - 321489*x + 184542631)
gp: K = bnfinit(x^6 - x^5 + 1699*x^4 - 1133*x^3 + 967309*x^2 - 321489*x + 184542631, 1)
Normalized defining polynomial
\( x^{6} - x^{5} + 1699 x^{4} - 1133 x^{3} + 967309 x^{2} - 321489 x + 184542631 \)
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: | \(-28121828974911=-\,3^{3}\cdot 7^{4}\cdot 757^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $174.38$ | 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: | $3, 7, 757$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(15897=3\cdot 7\cdot 757\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{15897}(2272,·)$, $\chi_{15897}(1,·)$, $\chi_{15897}(6812,·)$, $\chi_{15897}(9083,·)$, $\chi_{15897}(11356,·)$, $\chi_{15897}(2270,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{11784864013} a^{5} + \frac{840665152}{11784864013} a^{4} + \frac{3367846458}{11784864013} a^{3} + \frac{2110733863}{11784864013} a^{2} - \frac{5471781350}{11784864013} a + \frac{177795749}{906528001}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{2}\times C_{6}\times C_{924}$, which has order $11088$ (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: | \( \frac{4542}{11784864013} a^{5} + \frac{5180172}{11784864013} a^{4} + \frac{5123362}{11784864013} a^{3} + \frac{5858763177}{11784864013} a^{2} + \frac{1447311717}{11784864013} a + \frac{129465347210}{906528001} \), \( \frac{9098}{11784864013} a^{5} - \frac{5191541}{11784864013} a^{4} + \frac{20641084}{11784864013} a^{3} - \frac{5871655616}{11784864013} a^{2} + \frac{8783732625}{11784864013} a - \frac{129293733525}{906528001} \) (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 2.10181872849 \) (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
| \(\Q(\sqrt{-2271}) \), \(\Q(\zeta_{7})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | \(\Q\) $\times$ \(\Q(\sqrt{-2271}) \) $\times$ \(\Q(\zeta_{7})^+\) |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ | R | ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }$ | ${\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }$ | ${\href{/LocalNumberField/37.6.0.1}{6} }$ | ${\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }$ | ${\href{/LocalNumberField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| $7$ | 7.3.2.2 | $x^{3} - 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 7.3.2.2 | $x^{3} - 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 757 | Data not computed | ||||||