magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![78, -80, -28, -20, -14, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 14*x^4 - 20*x^3 - 28*x^2 - 80*x + 78)
gp: K = bnfinit(x^6 - 14*x^4 - 20*x^3 - 28*x^2 - 80*x + 78, 1)
Normalized defining polynomial
\( x^{6} - 14 x^{4} - 20 x^{3} - 28 x^{2} - 80 x + 78 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $6$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[2, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1703680000=2^{11}\cdot 5^{4}\cdot 11^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $34.56$ | 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: | $2, 5, 11$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2943} a^{5} + \frac{1057}{2943} a^{4} - \frac{1105}{2943} a^{3} + \frac{122}{981} a^{2} + \frac{1301}{2943} a + \frac{232}{981}$
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: | \( \frac{13}{327} a^{5} + \frac{7}{327} a^{4} - \frac{304}{327} a^{3} - \frac{49}{109} a^{2} + \frac{890}{327} a - \frac{145}{109} \), \( \frac{40}{327} a^{5} - \frac{230}{327} a^{4} + \frac{272}{327} a^{3} - \frac{243}{109} a^{2} + \frac{374}{327} a + \frac{15}{109} \), \( \frac{126}{109} a^{5} + \frac{420}{109} a^{4} - \frac{1890}{109} a^{3} - \frac{8820}{109} a^{2} - \frac{5460}{109} a - \frac{34493}{109} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 974.593056844 \) | 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 36 |
| The 9 conjugacy class representatives for $S_3^2$ |
| Character table for $S_3^2$ |
Intermediate fields
| \(\Q(\sqrt{22}) \) |
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: | 3.1.200.1 $\times$ 3.1.44.1 |
| Degree 9 sibling: | data not computed |
| Degree 12 sibling: | data not computed |
| Degree 18 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 | R | ${\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{3}$ | R | ${\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }$ | ${\href{/LocalNumberField/19.6.0.1}{6} }$ | ${\href{/LocalNumberField/23.6.0.1}{6} }$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }$ | ${\href{/LocalNumberField/37.6.0.1}{6} }$ | ${\href{/LocalNumberField/41.6.0.1}{6} }$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.6.11.13 | $x^{6} + 10$ | $6$ | $1$ | $11$ | $D_{6}$ | $[3]_{3}^{2}$ |
| $5$ | 5.6.4.2 | $x^{6} - 5 x^{3} + 50$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
| $11$ | 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 | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| * | 1.2e3_11.2t1.1c1 | $1$ | $ 2^{3} \cdot 11 $ | $x^{2} - 22$ | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.11.2t1.1c1 | $1$ | $ 11 $ | $x^{2} - x + 3$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.2e3.2t1.2c1 | $1$ | $ 2^{3}$ | $x^{2} + 2$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.2e3_5e2_11e2.6t3.2c1 | $2$ | $ 2^{3} \cdot 5^{2} \cdot 11^{2}$ | $x^{6} - x^{5} + 12 x^{4} - 13 x^{3} + 30 x^{2} + 25 x + 5$ | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| 2.2e3_5e2.3t2.1c1 | $2$ | $ 2^{3} \cdot 5^{2}$ | $x^{3} - x^{2} + 2 x + 2$ | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.2e2_11.3t2.1c1 | $2$ | $ 2^{2} \cdot 11 $ | $x^{3} - x^{2} + x + 1$ | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.2e6_11.6t3.3c1 | $2$ | $ 2^{6} \cdot 11 $ | $x^{6} - 6 x^{4} + 20 x^{2} - 22$ | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 4.2e8_5e4_11e2.6t9.1c1 | $4$ | $ 2^{8} \cdot 5^{4} \cdot 11^{2}$ | $x^{6} - 14 x^{4} - 20 x^{3} - 28 x^{2} - 80 x + 78$ | $S_3^2$ (as 6T9) | $1$ | $0$ |
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 *.