magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 1, 5, 4, -1, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - x^4 + 4*x^3 + 5*x^2 + x + 16)
gp: K = bnfinit(x^6 - 2*x^5 - x^4 + 4*x^3 + 5*x^2 + x + 16, 1)
Normalized defining polynomial
\( x^{6} - 2 x^{5} - x^{4} + 4 x^{3} + 5 x^{2} + x + 16 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $6$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 3]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-6040683=-\,3^{3}\cdot 11^{2}\cdot 43^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $13.50$ | 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: | $3, 11, 43$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{177} a^{5} - \frac{22}{59} a^{4} - \frac{25}{177} a^{3} + \frac{11}{177} a^{2} + \frac{3}{59} a - \frac{44}{177}$
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: | $2$ | 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: | \( \frac{1}{59} a^{5} + \frac{52}{59} a^{4} - \frac{84}{59} a^{3} - \frac{166}{59} a^{2} + \frac{245}{59} a + \frac{605}{59} \), \( \frac{38}{59} a^{5} + \frac{29}{59} a^{4} - \frac{242}{59} a^{3} - \frac{113}{59} a^{2} + \frac{814}{59} a + \frac{1101}{59} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 49.643382747 \) | 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
| 3.1.1419.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 4.2.12771.1 $\times$ \(\Q(\sqrt{473}) \) |
| Degree 6 sibling: | 6.2.8571729177.1 |
| Degree 8 siblings: | 8.4.36489851106489.1, 8.0.163098441.1 |
| Degree 12 siblings: | Deg 12, Deg 12, Deg 12, Deg 12, Deg 12, Deg 12 |
| Degree 16 sibling: | Deg 16 |
| 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 | ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/5.6.0.1}{6} }$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ | R | ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | ${\href{/LocalNumberField/17.6.0.1}{6} }$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ | ${\href{/LocalNumberField/41.6.0.1}{6} }$ | R | ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 3.4.3.2 | $x^{4} - 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| $11$ | 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $43$ | $\Q_{43}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{43}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 43.2.1.1 | $x^{2} - 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.2.1.1 | $x^{2} - 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
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_11_43.2t1.1c1 | $1$ | $ 3 \cdot 11 \cdot 43 $ | $x^{2} - x + 355$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.11_43.2t1.1c1 | $1$ | $ 11 \cdot 43 $ | $x^{2} - x - 118$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 2.3_11_43.6t3.5c1 | $2$ | $ 3 \cdot 11 \cdot 43 $ | $x^{6} - 27 x^{3} + 64$ | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.3_11_43.3t2.1c1 | $2$ | $ 3 \cdot 11 \cdot 43 $ | $x^{3} - x^{2} + 10 x - 12$ | $S_3$ (as 3T2) | $1$ | $0$ |
| 3.3e3_11_43.4t5.1c1 | $3$ | $ 3^{3} \cdot 11 \cdot 43 $ | $x^{4} - x^{3} - 3 x^{2} + 5 x + 1$ | $S_4$ (as 4T5) | $1$ | $1$ | |
| 3.3e3_11e2_43e2.6t11.2c1 | $3$ | $ 3^{3} \cdot 11^{2} \cdot 43^{2}$ | $x^{6} - 2 x^{5} - x^{4} + 4 x^{3} + 5 x^{2} + x + 16$ | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
| * | 3.3e2_11_43.6t11.2c1 | $3$ | $ 3^{2} \cdot 11 \cdot 43 $ | $x^{6} - 2 x^{5} - x^{4} + 4 x^{3} + 5 x^{2} + x + 16$ | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ |
| 3.3e2_11e2_43e2.6t8.2c1 | $3$ | $ 3^{2} \cdot 11^{2} \cdot 43^{2}$ | $x^{4} - x^{3} - 3 x^{2} + 5 x + 1$ | $S_4$ (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 *.