magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![181, -90, 81, -10, 18, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 + 18*x^4 - 10*x^3 + 81*x^2 - 90*x + 181)
gp: K = bnfinit(x^6 + 18*x^4 - 10*x^3 + 81*x^2 - 90*x + 181, 1)
Normalized defining polynomial
\( x^{6} + 18 x^{4} - 10 x^{3} + 81 x^{2} - 90 x + 181 \)
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: | \(-43243551=-\,3^{9}\cdot 13^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $18.73$ | 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: | $3, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(117=3^{2}\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{117}(1,·)$, $\chi_{117}(116,·)$, $\chi_{117}(38,·)$, $\chi_{117}(40,·)$, $\chi_{117}(77,·)$, $\chi_{117}(79,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{4} a^{3} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{508} a^{5} - \frac{39}{508} a^{4} + \frac{15}{508} a^{3} - \frac{107}{254} a^{2} - \frac{209}{508} a - \frac{97}{254}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{4}$, which has order $4$
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: | \( \frac{7}{508} a^{5} - \frac{19}{508} a^{4} + \frac{105}{508} a^{3} - \frac{57}{127} a^{2} + \frac{315}{508} a - \frac{171}{254} \), \( \frac{3}{508} a^{5} + \frac{5}{254} a^{4} + \frac{45}{508} a^{3} - \frac{7}{508} a^{2} + \frac{2}{127} a - \frac{37}{254} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 3.39714980258 \) | 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
| \(\Q(\sqrt{-39}) \), \(\Q(\zeta_{9})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | \(\Q(\zeta_{9})^+\) $\times$ \(\Q(\sqrt{-39}) \) $\times$ \(\Q\) |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ | R | ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/7.6.0.1}{6} }$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ | R | ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }$ | ${\href{/LocalNumberField/29.6.0.1}{6} }$ | ${\href{/LocalNumberField/31.6.0.1}{6} }$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.6.9.10 | $x^{6} + 6 x^{4} + 3$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ |
| $13$ | 13.6.3.1 | $x^{6} - 52 x^{4} + 676 x^{2} - 79092$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
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_13.2t1.1c1 | $1$ | $ 3 \cdot 13 $ | $x^{2} - x + 10$ | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.3e2.3t1.1c1 | $1$ | $ 3^{2}$ | $x^{3} - 3 x - 1$ | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.3e2_13.6t1.3c1 | $1$ | $ 3^{2} \cdot 13 $ | $x^{6} + 18 x^{4} - 10 x^{3} + 81 x^{2} - 90 x + 181$ | $C_6$ (as 6T1) | $0$ | $-1$ |
| * | 1.3e2.3t1.1c2 | $1$ | $ 3^{2}$ | $x^{3} - 3 x - 1$ | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.3e2_13.6t1.3c2 | $1$ | $ 3^{2} \cdot 13 $ | $x^{6} + 18 x^{4} - 10 x^{3} + 81 x^{2} - 90 x + 181$ | $C_6$ (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 *.