magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-64, -192, 996, -273, -435, 141, 63, -21, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 - 21*x^7 + 63*x^6 + 141*x^5 - 435*x^4 - 273*x^3 + 996*x^2 - 192*x - 64)
gp: K = bnfinit(x^9 - 3*x^8 - 21*x^7 + 63*x^6 + 141*x^5 - 435*x^4 - 273*x^3 + 996*x^2 - 192*x - 64, 1)
Normalized defining polynomial
\( x^{9} - 3 x^{8} - 21 x^{7} + 63 x^{6} + 141 x^{5} - 435 x^{4} - 273 x^{3} + 996 x^{2} - 192 x - 64 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $9$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[9, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(80676485676081=3^{16}\cdot 37^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $35.09$ | 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: | $3, 37$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{15856} a^{8} + \frac{9}{15856} a^{7} + \frac{87}{15856} a^{6} + \frac{1107}{15856} a^{5} - \frac{2431}{15856} a^{4} + \frac{2105}{15856} a^{3} - \frac{6725}{15856} a^{2} - \frac{53}{1982} a - \frac{330}{991}$
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: | $8$ | 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: | \( 16038.8573535 \) | 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 27 |
| The 11 conjugacy class representatives for $C_3^2:C_3$ |
| Character table for $C_3^2:C_3$ |
Intermediate fields
| \(\Q(\zeta_{9})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 9 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 | ${\href{/LocalNumberField/2.3.0.1}{3} }^{3}$ | R | ${\href{/LocalNumberField/5.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ | R | ${\href{/LocalNumberField/41.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.9.16.1 | $x^{9} + 3 x^{8} + 3 x^{6} + 3$ | $9$ | $1$ | $16$ | $C_3^2:C_3$ | $[2, 2]^{3}$ |
| $37$ | 37.3.2.2 | $x^{3} + 74$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 37.3.0.1 | $x^{3} - x + 2$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 37.3.2.3 | $x^{3} - 148$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
Artin representations
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 *.