magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6, -12, -11, -4, -4, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 4*x^4 - 4*x^3 - 11*x^2 - 12*x - 6)
gp: K = bnfinit(x^6 - 4*x^4 - 4*x^3 - 11*x^2 - 12*x - 6, 1)
Normalized defining polynomial
\( x^{6} - 4 x^{4} - 4 x^{3} - 11 x^{2} - 12 x - 6 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $6$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[2, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(6144000=2^{14}\cdot 3\cdot 5^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $13.53$ | 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: | $2, 3, 5$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a - \frac{1}{4}$
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: | $3$ | 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: | \( \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{5}{8} a^{3} - \frac{5}{8} a^{2} - \frac{5}{4} a - \frac{11}{4} \), \( \frac{7}{8} a^{5} - \frac{5}{8} a^{4} - \frac{27}{8} a^{3} - \frac{7}{8} a^{2} - \frac{31}{4} a - \frac{17}{4} \), \( \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - 2 a^{3} - \frac{1}{4} a^{2} - 5 a - \frac{5}{2} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 25.7070308797 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$S_3\wr C_2$ (as 6T13):
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 $C_3^2:D_4$ |
| Character table for $C_3^2:D_4$ |
Intermediate fields
| \(\Q(\sqrt{10}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 6.2.2211840.1 |
| Degree 6 sibling: | 6.2.2211840.1 |
| Degree 9 sibling: | 9.1.14155776000.1 |
| Degree 12 siblings: | Deg 12, Deg 12, Deg 12, Deg 12, Deg 12, Deg 12 |
| Degree 18 siblings: | Deg 18, Deg 18, Deg 18 |
| 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 | R | ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ | ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | ${\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ | ${\href{/LocalNumberField/19.6.0.1}{6} }$ | ${\href{/LocalNumberField/23.6.0.1}{6} }$ | ${\href{/LocalNumberField/29.6.0.1}{6} }$ | ${\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }$ | ${\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.3.2 | $x^{2} + 6$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.4.11.20 | $x^{4} + 6$ | $4$ | $1$ | $11$ | $D_{4}$ | $[2, 3, 4]$ | |
| $3$ | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| $5$ | 5.6.3.2 | $x^{6} - 25 x^{2} + 250$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.2e2_3_5.2t1.1c1 | $1$ | $ 2^{2} \cdot 3 \cdot 5 $ | $x^{2} - 15$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| * | 1.2e3_5.2t1.1c1 | $1$ | $ 2^{3} \cdot 5 $ | $x^{2} - 10$ | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.2e3_3.2t1.1c1 | $1$ | $ 2^{3} \cdot 3 $ | $x^{2} - 6$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 2.2e8_3_5.4t3.14c1 | $2$ | $ 2^{8} \cdot 3 \cdot 5 $ | $x^{4} + 16 x^{2} + 40$ | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
| 4.2e13_3e3_5e2.12t34.2c1 | $4$ | $ 2^{13} \cdot 3^{3} \cdot 5^{2}$ | $x^{6} - 4 x^{4} - 4 x^{3} - 11 x^{2} - 12 x - 6$ | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| * | 4.2e11_3_5e2.6t13.1c1 | $4$ | $ 2^{11} \cdot 3 \cdot 5^{2}$ | $x^{6} - 4 x^{4} - 4 x^{3} - 11 x^{2} - 12 x - 6$ | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ |
| 4.2e11_3e2_5.6t13.1c1 | $4$ | $ 2^{11} \cdot 3^{2} \cdot 5 $ | $x^{6} - 4 x^{4} - 4 x^{3} - 11 x^{2} - 12 x - 6$ | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| 4.2e13_3e2_5e3.12t34.2c1 | $4$ | $ 2^{13} \cdot 3^{2} \cdot 5^{3}$ | $x^{6} - 4 x^{4} - 4 x^{3} - 11 x^{2} - 12 x - 6$ | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.