magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6, 24, 32, 43, 33, 22, 5, 0, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 + 5*x^6 + 22*x^5 + 33*x^4 + 43*x^3 + 32*x^2 + 24*x - 6)
gp: K = bnfinit(x^9 - x^8 + 5*x^6 + 22*x^5 + 33*x^4 + 43*x^3 + 32*x^2 + 24*x - 6, 1)
Normalized defining polynomial
\( x^{9} - x^{8} + 5 x^{6} + 22 x^{5} + 33 x^{4} + 43 x^{3} + 32 x^{2} + 24 x - 6 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $9$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[1, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1826195471424=2^{6}\cdot 3^{4}\cdot 137^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $23.04$ | 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: | $2, 3, 137$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{4}{9} a^{3} - \frac{4}{9} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{27} a^{7} - \frac{4}{27} a^{5} + \frac{1}{27} a^{4} + \frac{4}{9} a^{3} - \frac{13}{27} a^{2} + \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{81} a^{8} + \frac{1}{81} a^{7} - \frac{4}{81} a^{6} - \frac{4}{27} a^{5} - \frac{23}{81} a^{4} + \frac{8}{81} a^{3} + \frac{5}{81} a^{2} - \frac{11}{27} a + \frac{4}{27}$
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: | $4$ | 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: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 3089.69538974 \) | 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 18 |
| The 6 conjugacy class representatives for $D_{9}$ |
| Character table for $D_{9}$ |
Intermediate fields
| 3.1.411.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | 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 | R | ${\href{/LocalNumberField/5.9.0.1}{9} }$ | ${\href{/LocalNumberField/7.9.0.1}{9} }$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | ${\href{/LocalNumberField/19.9.0.1}{9} }$ | ${\href{/LocalNumberField/23.9.0.1}{9} }$ | ${\href{/LocalNumberField/29.9.0.1}{9} }$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.9.0.1}{9} }$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.9.0.1}{9} }$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| $3$ | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 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 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $137$ | $\Q_{137}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 137.2.1.2 | $x^{2} + 411$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 137.2.1.2 | $x^{2} + 411$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 137.2.1.2 | $x^{2} + 411$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 137.2.1.2 | $x^{2} + 411$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
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_137.2t1.1c1 | $1$ | $ 3 \cdot 137 $ | $x^{2} - x + 103$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 2.3_137.3t2.1c1 | $2$ | $ 3 \cdot 137 $ | $x^{3} - x^{2} + 5 x - 2$ | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.2e2_3_137.9t3.1c1 | $2$ | $ 2^{2} \cdot 3 \cdot 137 $ | $x^{9} - x^{8} + 5 x^{6} + 22 x^{5} + 33 x^{4} + 43 x^{3} + 32 x^{2} + 24 x - 6$ | $D_{9}$ (as 9T3) | $1$ | $0$ |
| * | 2.2e2_3_137.9t3.1c2 | $2$ | $ 2^{2} \cdot 3 \cdot 137 $ | $x^{9} - x^{8} + 5 x^{6} + 22 x^{5} + 33 x^{4} + 43 x^{3} + 32 x^{2} + 24 x - 6$ | $D_{9}$ (as 9T3) | $1$ | $0$ |
| * | 2.2e2_3_137.9t3.1c3 | $2$ | $ 2^{2} \cdot 3 \cdot 137 $ | $x^{9} - x^{8} + 5 x^{6} + 22 x^{5} + 33 x^{4} + 43 x^{3} + 32 x^{2} + 24 x - 6$ | $D_{9}$ (as 9T3) | $1$ | $0$ |
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 *.