magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![285768000, 0, 95256000, 0, 11113200, 0, 604800, 0, 16380, 0, 210, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 210*x^10 + 16380*x^8 + 604800*x^6 + 11113200*x^4 + 95256000*x^2 + 285768000)
gp: K = bnfinit(x^12 + 210*x^10 + 16380*x^8 + 604800*x^6 + 11113200*x^4 + 95256000*x^2 + 285768000, 1)
Normalized defining polynomial
\( x^{12} + 210 x^{10} + 16380 x^{8} + 604800 x^{6} + 11113200 x^{4} + 95256000 x^{2} + 285768000 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $12$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(105433321738752000000000=2^{18}\cdot 3^{6}\cdot 5^{9}\cdot 7^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $82.91$ | 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: | $2, 3, 5, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(840=2^{3}\cdot 3\cdot 5\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(293,·)$, $\chi_{840}(289,·)$, $\chi_{840}(169,·)$, $\chi_{840}(173,·)$, $\chi_{840}(529,·)$, $\chi_{840}(773,·)$, $\chi_{840}(437,·)$, $\chi_{840}(361,·)$, $\chi_{840}(121,·)$, $\chi_{840}(797,·)$, $\chi_{840}(677,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{6} a^{2}$, $\frac{1}{6} a^{3}$, $\frac{1}{180} a^{4}$, $\frac{1}{180} a^{5}$, $\frac{1}{7560} a^{6}$, $\frac{1}{7560} a^{7}$, $\frac{1}{226800} a^{8}$, $\frac{1}{226800} a^{9}$, $\frac{1}{1360800} a^{10}$, $\frac{1}{1360800} a^{11}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{2}\times C_{2}\times C_{3970}$, which has order $15880$ (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: | $5$ | 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: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 104.882003477 \) (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 12 |
| The 12 conjugacy class representatives for $C_{12}$ |
| Character table for $C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.3528000.1, 6.6.300125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | R | ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/17.12.0.1}{12} }$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/23.12.0.1}{12} }$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/37.12.0.1}{12} }$ | ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }$ | ${\href{/LocalNumberField/53.12.0.1}{12} }$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.12.18.28 | $x^{12} - 52 x^{10} + 1100 x^{8} - 12000 x^{6} - 61072 x^{4} + 62144 x^{2} - 62144$ | $2$ | $6$ | $18$ | $C_{12}$ | $[3]^{6}$ |
| $3$ | 3.12.6.1 | $x^{12} - 243 x^{2} + 1458$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |
| $5$ | 5.12.9.1 | $x^{12} - 10 x^{8} - 375 x^{4} - 2000$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
| $7$ | 7.12.10.5 | $x^{12} + 56 x^{6} + 1323$ | $6$ | $2$ | $10$ | $C_{12}$ | $[\ ]_{6}^{2}$ |