magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![72, 0, -432, 0, 756, 0, -552, 0, 180, 0, -24, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 24*x^10 + 180*x^8 - 552*x^6 + 756*x^4 - 432*x^2 + 72)
gp: K = bnfinit(x^12 - 24*x^10 + 180*x^8 - 552*x^6 + 756*x^4 - 432*x^2 + 72, 1)
Normalized defining polynomial
\( x^{12} - 24 x^{10} + 180 x^{8} - 552 x^{6} + 756 x^{4} - 432 x^{2} + 72 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $12$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[12, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(3327916660110655488=2^{33}\cdot 3^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $34.96$ | 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: | $2, 3$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(144=2^{4}\cdot 3^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{144}(1,·)$, $\chi_{144}(83,·)$, $\chi_{144}(131,·)$, $\chi_{144}(97,·)$, $\chi_{144}(73,·)$, $\chi_{144}(107,·)$, $\chi_{144}(11,·)$, $\chi_{144}(49,·)$, $\chi_{144}(35,·)$, $\chi_{144}(25,·)$, $\chi_{144}(121,·)$, $\chi_{144}(59,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{6} a^{6}$, $\frac{1}{6} a^{7}$, $\frac{1}{12} a^{8}$, $\frac{1}{12} a^{9}$, $\frac{1}{204} a^{10} + \frac{7}{204} a^{8} + \frac{1}{34} a^{6} + \frac{7}{34} a^{4} - \frac{7}{17} a^{2} + \frac{2}{17}$, $\frac{1}{204} a^{11} + \frac{7}{204} a^{9} + \frac{1}{34} a^{7} + \frac{7}{34} a^{5} - \frac{7}{17} a^{3} + \frac{2}{17} a$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
Trivial group, which has order $1$
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $11$ | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 153152.667026 \) | 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
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{9})^+\), 4.4.18432.1, 6.6.3359232.1 |
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 | ${\href{/LocalNumberField/5.12.0.1}{12} }$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/11.12.0.1}{12} }$ | ${\href{/LocalNumberField/13.12.0.1}{12} }$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/29.12.0.1}{12} }$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/43.12.0.1}{12} }$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.12.33.376 | $x^{12} + 36 x^{10} + 42 x^{8} - 40 x^{6} + 40 x^{4} + 32 x^{2} - 56$ | $4$ | $3$ | $33$ | $C_{12}$ | $[3, 4]^{3}$ |
| $3$ | 3.12.18.74 | $x^{12} - 6 x^{11} - 3 x^{10} - 12 x^{9} + 9 x^{8} + 9 x^{7} + 12 x^{6} - 9 x^{3} - 9$ | $6$ | $2$ | $18$ | $C_{12}$ | $[2]_{2}^{2}$ |