magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![78, 27, 9, -9, -6, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 6*x^4 - 9*x^3 + 9*x^2 + 27*x + 78)
gp: K = bnfinit(x^6 - 6*x^4 - 9*x^3 + 9*x^2 + 27*x + 78, 1)
Normalized defining polynomial
\( x^{6} - 6 x^{4} - 9 x^{3} + 9 x^{2} + 27 x + 78 \)
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: | \(-8985939039=-\,3^{9}\cdot 7^{3}\cdot 11^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $45.60$ | 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: | $3, 7, 11$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{80} a^{5} - \frac{9}{80} a^{4} - \frac{1}{16} a^{3} + \frac{9}{20} a^{2} + \frac{1}{16} a - \frac{9}{40}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{3}\times C_{6}\times C_{6}$, which has order $108$
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: | \( \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{5}{8} a^{3} + \frac{1}{2} a^{2} - \frac{3}{8} a + \frac{19}{4} \), \( \frac{1}{40} a^{5} - \frac{9}{40} a^{4} - \frac{1}{8} a^{3} + \frac{19}{10} a^{2} + \frac{9}{8} a - \frac{149}{20} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 39.2068104305 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3\times S_3$ (as 6T5):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| 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
| \(\Q(\sqrt{-231}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Galois closure: | data not computed |
| Twin sextic algebra: | 3.1.231.1 $\times$ \(\Q(\zeta_{9})^+\) |
| Degree 9 sibling: | 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 | ${\href{/LocalNumberField/2.3.0.1}{3} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{3}$ | R | ${\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{3}$ | R | R | ${\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }$ | ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }$ | ${\href{/LocalNumberField/43.6.0.1}{6} }$ | ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.6.9.3 | $x^{6} + 3 x^{4} + 24$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ |
| $7$ | 7.6.3.1 | $x^{6} - 14 x^{4} + 49 x^{2} - 1372$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| $11$ | 11.6.3.1 | $x^{6} - 22 x^{4} + 121 x^{2} - 11979$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| * | 1.3_7_11.2t1.1c1 | $1$ | $ 3 \cdot 7 \cdot 11 $ | $x^{2} - x + 58$ | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.3e2.3t1.1c1 | $1$ | $ 3^{2}$ | $x^{3} - 3 x - 1$ | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.3e2_7_11.6t1.2c1 | $1$ | $ 3^{2} \cdot 7 \cdot 11 $ | $x^{6} + 114 x^{4} - 58 x^{3} + 3249 x^{2} - 3306 x + 23941$ | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.3e2_7_11.6t1.2c2 | $1$ | $ 3^{2} \cdot 7 \cdot 11 $ | $x^{6} + 114 x^{4} - 58 x^{3} + 3249 x^{2} - 3306 x + 23941$ | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.3e2.3t1.1c2 | $1$ | $ 3^{2}$ | $x^{3} - 3 x - 1$ | $C_3$ (as 3T1) | $0$ | $1$ | |
| 2.3_7_11.3t2.1c1 | $2$ | $ 3 \cdot 7 \cdot 11 $ | $x^{3} - x^{2} + 3$ | $S_3$ (as 3T2) | $1$ | $0$ | |
| * | 2.3e4_7_11.6t5.2c1 | $2$ | $ 3^{4} \cdot 7 \cdot 11 $ | $x^{6} - 6 x^{4} - 9 x^{3} + 9 x^{2} + 27 x + 78$ | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.3e4_7_11.6t5.2c2 | $2$ | $ 3^{4} \cdot 7 \cdot 11 $ | $x^{6} - 6 x^{4} - 9 x^{3} + 9 x^{2} + 27 x + 78$ | $S_3\times C_3$ (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 *.