magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-56, 36, -8, -9, 0, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 9*x^3 - 8*x^2 + 36*x - 56)
gp: K = bnfinit(x^6 - x^5 - 9*x^3 - 8*x^2 + 36*x - 56, 1)
Normalized defining polynomial
\( x^{6} - x^{5} - 9 x^{3} - 8 x^{2} + 36 x - 56 \)
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: | \(2031889=23^{3}\cdot 167\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $11.25$ | 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: | $23, 167$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{3316} a^{5} - \frac{37}{3316} a^{4} + \frac{333}{829} a^{3} - \frac{1537}{3316} a^{2} - \frac{262}{829} a + \frac{322}{829}$
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: | $3$ | 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: | \( \frac{41}{3316} a^{5} + \frac{141}{3316} a^{4} - \frac{51}{1658} a^{3} - \frac{13}{3316} a^{2} - \frac{759}{1658} a - \frac{62}{829} \), \( \frac{23}{1658} a^{5} - \frac{11}{829} a^{4} - \frac{37}{1658} a^{3} - \frac{533}{1658} a^{2} + \frac{1595}{1658} a - \frac{939}{829} \), \( \frac{117}{3316} a^{5} + \frac{645}{3316} a^{4} - \frac{833}{1658} a^{3} - \frac{765}{3316} a^{2} + \frac{2525}{1658} a - \frac{1289}{829} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 6.9929779899 \) | 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
| 3.1.23.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 4.2.641447.1 $\times$ \(\Q(\sqrt{-167}) \) |
| Degree 6 sibling: | 6.0.88343.1 |
| Degree 8 siblings: | 8.4.217659300264961.1, 8.0.411454253809.1 |
| 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 | ${\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ | ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }$ | ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ | R | ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $23$ | 23.2.1.1 | $x^{2} - 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $167$ | $\Q_{167}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{167}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 167.2.1.1 | $x^{2} - 167$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 167.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.167.2t1.1c1 | $1$ | $ 167 $ | $x^{2} - x + 42$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.23.2t1.1c1 | $1$ | $ 23 $ | $x^{2} - x + 6$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.23_167.2t1.1c1 | $1$ | $ 23 \cdot 167 $ | $x^{2} - x - 960$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 2.23_167e2.6t3.2c1 | $2$ | $ 23 \cdot 167^{2}$ | $x^{6} - x^{5} + 125 x^{4} - 85 x^{3} + 5251 x^{2} - 1513 x + 74047$ | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.23.3t2.1c1 | $2$ | $ 23 $ | $x^{3} - x^{2} + 1$ | $S_3$ (as 3T2) | $1$ | $0$ |
| 3.23_167e2.4t5.1c1 | $3$ | $ 23 \cdot 167^{2}$ | $x^{4} - x^{3} - 16 x^{2} + 29 x + 6$ | $S_4$ (as 4T5) | $1$ | $1$ | |
| * | 3.23e2_167.6t11.1c1 | $3$ | $ 23^{2} \cdot 167 $ | $x^{6} - x^{5} - 9 x^{3} - 8 x^{2} + 36 x - 56$ | $S_4\times C_2$ (as 6T11) | $1$ | $1$ |
| 3.23_167.6t11.1c1 | $3$ | $ 23 \cdot 167 $ | $x^{6} - x^{5} - 9 x^{3} - 8 x^{2} + 36 x - 56$ | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
| 3.23e2_167e2.6t8.1c1 | $3$ | $ 23^{2} \cdot 167^{2}$ | $x^{4} - x^{3} - 16 x^{2} + 29 x + 6$ | $S_4$ (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 *.