magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -3, -2, -3, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^4 - 2*x^3 - 3*x^2 + 1)
gp: K = bnfinit(x^6 - 3*x^4 - 2*x^3 - 3*x^2 + 1, 1)
Normalized defining polynomial
\( x^{6} - 3 x^{4} - 2 x^{3} - 3 x^{2} + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $6$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 1]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-6858432=-\,2^{6}\cdot 3^{7}\cdot 7^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $13.78$ | 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: | $2, 3, 7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$
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: | $4$ | 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: | \( a \), \( a^{5} - 3 a^{3} - 2 a^{2} - 2 a + 1 \), \( a^{5} - 3 a^{3} - 2 a^{2} - 2 a - 1 \), \( a^{5} - 2 a^{4} - 3 a^{3} + 6 a^{2} - 1 \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 39.0307722518 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times S_4$ (as 6T11):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 48 |
| The 10 conjugacy class representatives for $S_4\times C_2$ |
| Character table for $S_4\times C_2$ |
Intermediate fields
| 3.3.756.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 4.0.3024.1 $\times$ \(\Q(\sqrt{-7}) \) |
| Degree 6 sibling: | 6.4.16003008.3 |
| Degree 8 siblings: | 8.0.448084224.9, 8.0.82301184.1 |
| Degree 12 siblings: | 12.4.439022168653824.1, 12.0.47038089498624.1, 12.0.2304866385432576.1, 12.8.2304866385432576.1, 12.0.2304866385432576.5, 12.0.256096265048064.2 |
| Degree 16 sibling: | 16.0.16263137215612256256.1 |
| Degree 24 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 | ${\href{/LocalNumberField/5.6.0.1}{6} }$ | R | ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | ${\href{/LocalNumberField/17.6.0.1}{6} }$ | ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.6.6.8 | $x^{6} + 2 x + 2$ | $6$ | $1$ | $6$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ |
| $3$ | 3.6.7.5 | $x^{6} + 6 x^{2} + 3$ | $6$ | $1$ | $7$ | $D_{6}$ | $[3/2]_{2}^{2}$ |
| $7$ | $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.3.2t1.1c1 | $1$ | $ 3 $ | $x^{2} - x + 1$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.3_7.2t1.1c1 | $1$ | $ 3 \cdot 7 $ | $x^{2} - x - 5$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.7.2t1.1c1 | $1$ | $ 7 $ | $x^{2} - x + 2$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.2e2_3e3_7.6t3.2c1 | $2$ | $ 2^{2} \cdot 3^{3} \cdot 7 $ | $x^{6} - 2 x^{3} + 8$ | $D_{6}$ (as 6T3) | $1$ | $-2$ | |
| * | 2.2e2_3e3_7.3t2.2c1 | $2$ | $ 2^{2} \cdot 3^{3} \cdot 7 $ | $x^{3} - 6 x - 2$ | $S_3$ (as 3T2) | $1$ | $2$ |
| 3.2e4_3e3_7.4t5.1c1 | $3$ | $ 2^{4} \cdot 3^{3} \cdot 7 $ | $x^{4} - 2 x^{3} + 3$ | $S_4$ (as 4T5) | $1$ | $-1$ | |
| 3.2e4_3e3_7e2.6t11.5c1 | $3$ | $ 2^{4} \cdot 3^{3} \cdot 7^{2}$ | $x^{6} - 3 x^{4} - 2 x^{3} - 3 x^{2} + 1$ | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
| * | 3.2e4_3e4_7.6t11.3c1 | $3$ | $ 2^{4} \cdot 3^{4} \cdot 7 $ | $x^{6} - 3 x^{4} - 2 x^{3} - 3 x^{2} + 1$ | $S_4\times C_2$ (as 6T11) | $1$ | $1$ |
| 3.2e4_3e4_7e2.6t8.1c1 | $3$ | $ 2^{4} \cdot 3^{4} \cdot 7^{2}$ | $x^{4} - 2 x^{3} + 3$ | $S_4$ (as 4T5) | $1$ | $-1$ |
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 *.