magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, 0, 32, 0, 16, 0, 8, 0, 4, 0, 2, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 2*x^10 + 4*x^8 + 8*x^6 + 16*x^4 + 32*x^2 + 64)
gp: K = bnfinit(x^12 + 2*x^10 + 4*x^8 + 8*x^6 + 16*x^4 + 32*x^2 + 64, 1)
Normalized defining polynomial
\( x^{12} + 2 x^{10} + 4 x^{8} + 8 x^{6} + 16 x^{4} + 32 x^{2} + 64 \)
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: | \(74049191673856=2^{18}\cdot 7^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $14.32$ | 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: | $2, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(56=2^{3}\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{56}(1,·)$, $\chi_{56}(5,·)$, $\chi_{56}(33,·)$, $\chi_{56}(9,·)$, $\chi_{56}(13,·)$, $\chi_{56}(45,·)$, $\chi_{56}(17,·)$, $\chi_{56}(53,·)$, $\chi_{56}(41,·)$, $\chi_{56}(25,·)$, $\chi_{56}(29,·)$, $\chi_{56}(37,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{2}$, which has order $2$
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: | \( -\frac{1}{16} a^{8} \) (order $14$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | \( \frac{1}{2} a^{2} + 1 \), \( \frac{1}{16} a^{8} + \frac{1}{2} a^{2} \), \( \frac{1}{16} a^{9} + \frac{1}{2} a^{2} \), \( \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{2} a^{2} \), \( \frac{1}{16} a^{9} + \frac{1}{16} a^{8} + \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{1}{4} a^{5} - 1 \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 279.150027194 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_6$ (as 12T2):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| An abelian group of order 12 |
| The 12 conjugacy class representatives for $C_6\times C_2$ |
| Character table for $C_6\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-14}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\zeta_{7})\), 6.6.1229312.1, 6.0.8605184.1 |
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 | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.6.9.1 | $x^{6} + 4 x^{4} + 4 x^{2} - 8$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ |
| 2.6.9.1 | $x^{6} + 4 x^{4} + 4 x^{2} - 8$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
| $7$ | 7.6.5.5 | $x^{6} + 56$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 7.6.5.5 | $x^{6} + 56$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |