sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^4 - 2*x^3 + 3*x^2 + x + 1)
gp: K = bnfinit(x^6 - x^4 - 2*x^3 + 3*x^2 + x + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, 3, -2, -1, 0, 1]);
Normalized defining polynomial
\( x^{6} - x^{4} - 2 x^{3} + 3 x^{2} + x + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
| Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(-107811=-\,3^{4}\cdot 11^{3}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $6.90$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $3, 11$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Aut(K/\Q)|$: | $3$ | ||
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
Class group and class number
Trivial group, which has order $1$
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
Unit group
sage: UK = K.unit_group()
magma: UK, f := UnitGroup(K);
| Rank: | $2$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| Fundamental units: | \( a \), \( \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{4}{5} a^{3} - \frac{6}{5} a^{2} + \frac{3}{5} a + \frac{6}{5} \) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 1.3637383458 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Galois group
$C_3\times S_3$ (as 6T5):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
| A solvable group of order 18 |
| The 9 conjugacy class representatives for $S_3\times C_3$ |
| Character table for $S_3\times C_3$ |
Intermediate fields
| \(\Q(\sqrt{-11}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Galois closure: | 18.0.665954073415574215371.1 |
| Twin sextic algebra: | \(\Q(\zeta_{9})^+\) $\times$ 3.1.891.1 |
| Degree 9 sibling: | 9.3.707347971.1 |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.6.0.1}{6} }$ | R | ${\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/7.6.0.1}{6} }$ | R | ${\href{/LocalNumberField/13.6.0.1}{6} }$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }$ | ${\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }$ | ${\href{/LocalNumberField/43.6.0.1}{6} }$ | ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
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]])
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];
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $3$ | 3.3.4.3 | $x^{3} - 3 x^{2} + 12$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
| 3.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| $11$ | 11.6.3.2 | $x^{6} - 121 x^{2} + 3993$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| * | 1.11.2t1.a.a | $1$ | $ 11 $ | $x^{2} - x + 3$ | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.9.3t1.a.a | $1$ | $ 3^{2}$ | $x^{3} - 3 x - 1$ | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.99.6t1.a.a | $1$ | $ 3^{2} \cdot 11 $ | $x^{6} - 3 x^{5} + 6 x^{4} - 5 x^{3} + 33 x^{2} - 54 x + 111$ | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.99.6t1.a.b | $1$ | $ 3^{2} \cdot 11 $ | $x^{6} - 3 x^{5} + 6 x^{4} - 5 x^{3} + 33 x^{2} - 54 x + 111$ | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.9.3t1.a.b | $1$ | $ 3^{2}$ | $x^{3} - 3 x - 1$ | $C_3$ (as 3T1) | $0$ | $1$ | |
| 2.891.3t2.b.a | $2$ | $ 3^{4} \cdot 11 $ | $x^{3} + 6 x - 1$ | $S_3$ (as 3T2) | $1$ | $0$ | |
| * | 2.99.6t5.a.a | $2$ | $ 3^{2} \cdot 11 $ | $x^{6} - x^{4} - 2 x^{3} + 3 x^{2} + x + 1$ | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.99.6t5.a.b | $2$ | $ 3^{2} \cdot 11 $ | $x^{6} - x^{4} - 2 x^{3} + 3 x^{2} + x + 1$ | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
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 *.