magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, 0, 0, 160, 0, 0, 56, 0, 0, -16, 0, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 16*x^9 + 56*x^6 + 160*x^3 - 8)
gp: K = bnfinit(x^12 - 16*x^9 + 56*x^6 + 160*x^3 - 8, 1)
Normalized defining polynomial
\( x^{12} - 16 x^{9} + 56 x^{6} + 160 x^{3} - 8 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $12$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[2, 5]$ | 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 not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{452} a^{9} - \frac{17}{113} a^{6} - \frac{6}{113} a^{3} + \frac{13}{113}$, $\frac{1}{452} a^{10} - \frac{17}{113} a^{7} - \frac{6}{113} a^{4} + \frac{13}{113} a$, $\frac{1}{452} a^{11} + \frac{45}{452} a^{8} - \frac{6}{113} a^{5} + \frac{13}{113} a^{2}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{3}$, which has order $3$
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $6$ | 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: | \( 34178.8560928 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$S_3\wr C_2$ (as 12T36):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 72 |
| The 9 conjugacy class representatives for $S_3\wr C_2$ |
| Character table for $S_3\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{6}) \), 4.2.18432.1, 6.4.322486272.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 6 siblings: | data not computed |
| Degree 9 sibling: | data not computed |
| Degree 12 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | 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 | R | R | ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.11.14 | $x^{4} + 8 x + 10$ | $4$ | $1$ | $11$ | $D_{4}$ | $[3, 4]^{2}$ |
| 2.8.22.8 | $x^{8} + 12 x^{6} + 10 x^{4} + 8 x^{2} + 36$ | $4$ | $2$ | $22$ | $D_4$ | $[3, 4]^{2}$ | |
| $3$ | 3.12.18.85 | $x^{12} + 36 x^{11} + 111 x^{10} + 90 x^{9} + 36 x^{8} + 90 x^{7} + 30 x^{6} + 108 x^{5} - 36 x^{4} + 54 x^{3} - 81 x^{2} + 54 x - 18$ | $6$ | $2$ | $18$ | $D_6$ | $[2]_{2}^{2}$ |