magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![415, -118, 85, 1, 6, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 6*x^4 + x^3 + 85*x^2 - 118*x + 415)
gp: K = bnfinit(x^6 - x^5 + 6*x^4 + x^3 + 85*x^2 - 118*x + 415, 1)
Normalized defining polynomial
\( x^{6} - x^{5} + 6 x^{4} + x^{3} + 85 x^{2} - 118 x + 415 \)
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: | \(-195899899=-\,13^{4}\cdot 19^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $24.10$ | 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: | $13, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(247=13\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{247}(1,·)$, $\chi_{247}(113,·)$, $\chi_{247}(170,·)$, $\chi_{247}(172,·)$, $\chi_{247}(94,·)$, $\chi_{247}(191,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{3925} a^{5} - \frac{253}{3925} a^{4} + \frac{177}{3925} a^{3} + \frac{1712}{3925} a^{2} - \frac{374}{3925} a - \frac{171}{785}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{13}$, which has order $13$
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: | \( \frac{18}{3925} a^{5} + \frac{156}{3925} a^{4} + \frac{46}{3925} a^{3} + \frac{201}{3925} a^{2} + \frac{1903}{3925} a + \frac{1632}{785} \), \( \frac{28}{3925} a^{5} - \frac{19}{3925} a^{4} + \frac{246}{3925} a^{3} + \frac{51}{3925} a^{2} + \frac{518}{3925} a + \frac{707}{785} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 5.46019947038 \) | 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 cyclic group of order 6 |
| The 6 conjugacy class representatives for $C_6$ |
| Character table for $C_6$ |
Intermediate fields
| \(\Q(\sqrt{-19}) \), 3.3.169.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 3.3.169.1 $\times$ \(\Q(\sqrt{-19}) \) $\times$ \(\Q\) |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.6.0.1}{6} }$ | ${\href{/LocalNumberField/3.6.0.1}{6} }$ | ${\href{/LocalNumberField/5.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ | R | ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ | R | ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }$ | ${\href{/LocalNumberField/41.6.0.1}{6} }$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $13$ | 13.6.4.3 | $x^{6} + 65 x^{3} + 1352$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| $19$ | 19.6.3.2 | $x^{6} - 361 x^{2} + 27436$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| * | 1.19.2t1.1c1 | $1$ | $ 19 $ | $x^{2} - x + 5$ | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.13.3t1.1c1 | $1$ | $ 13 $ | $x^{3} - x^{2} - 4 x - 1$ | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.13_19.6t1.3c1 | $1$ | $ 13 \cdot 19 $ | $x^{6} - x^{5} + 6 x^{4} + x^{3} + 85 x^{2} - 118 x + 415$ | $C_6$ (as 6T1) | $0$ | $-1$ |
| * | 1.13.3t1.1c2 | $1$ | $ 13 $ | $x^{3} - x^{2} - 4 x - 1$ | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.13_19.6t1.3c2 | $1$ | $ 13 \cdot 19 $ | $x^{6} - x^{5} + 6 x^{4} + x^{3} + 85 x^{2} - 118 x + 415$ | $C_6$ (as 6T1) | $0$ | $-1$ |
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 *.