magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![54, 18, 6, 3, -5, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 5*x^4 + 3*x^3 + 6*x^2 + 18*x + 54)
gp: K = bnfinit(x^6 - x^5 - 5*x^4 + 3*x^3 + 6*x^2 + 18*x + 54, 1)
Normalized defining polynomial
\( x^{6} - x^{5} - 5 x^{4} + 3 x^{3} + 6 x^{2} + 18 x + 54 \)
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: | \(-73926000=-\,2^{4}\cdot 3^{3}\cdot 5^{3}\cdot 37^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $20.49$ | 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: | $2, 3, 5, 37$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{4} + \frac{1}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{18} a^{5} - \frac{1}{18} a^{3} - \frac{1}{3} a$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{6}$, which has order $6$
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: | \( \frac{1}{18} a^{5} - \frac{7}{18} a^{3} - \frac{1}{3} a^{2} + \frac{5}{3} a + 2 \), \( \frac{2}{9} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} + \frac{4}{3} a^{2} - \frac{7}{3} a - 7 \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 39.5542979739 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 12 |
| The 6 conjugacy class representatives for $D_{6}$ |
| Character table for $D_{6}$ |
Intermediate fields
| \(\Q(\sqrt{-15}) \), 3.1.2220.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Galois closure: | Deg 12 |
| Twin sextic algebra: | 3.1.2220.1 $\times$ \(\Q(\sqrt{37}) \) $\times$ \(\Q\) |
| Degree 6 sibling: | 6.2.182350800.1 |
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 | R | ${\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.3.2.1 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.3.2.1 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| $3$ | 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $5$ | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $37$ | 37.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 37.4.2.1 | $x^{4} + 333 x^{2} + 34225$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
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_5_37.2t1.1c1 | $1$ | $ 3 \cdot 5 \cdot 37 $ | $x^{2} - x + 139$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.3_5.2t1.1c1 | $1$ | $ 3 \cdot 5 $ | $x^{2} - x + 4$ | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.37.2t1.1c1 | $1$ | $ 37 $ | $x^{2} - x - 9$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| * | 2.2e2_3_5_37.3t2.1c1 | $2$ | $ 2^{2} \cdot 3 \cdot 5 \cdot 37 $ | $x^{3} - x^{2} - 11 x + 21$ | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.2e2_3_5_37.6t3.6c1 | $2$ | $ 2^{2} \cdot 3 \cdot 5 \cdot 37 $ | $x^{6} - x^{5} - 5 x^{4} + 3 x^{3} + 6 x^{2} + 18 x + 54$ | $D_{6}$ (as 6T3) | $1$ | $0$ |
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 *.