magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-56, 84, -84, 21, -21, 21, -15, 3, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 3*x^7 - 15*x^6 + 21*x^5 - 21*x^4 + 21*x^3 - 84*x^2 + 84*x - 56)
gp: K = bnfinit(x^9 - 3*x^8 + 3*x^7 - 15*x^6 + 21*x^5 - 21*x^4 + 21*x^3 - 84*x^2 + 84*x - 56, 1)
Normalized defining polynomial
\( x^{9} - 3 x^{8} + 3 x^{7} - 15 x^{6} + 21 x^{5} - 21 x^{4} + 21 x^{3} - 84 x^{2} + 84 x - 56 \)
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: | \(5064403678929=3^{16}\cdot 7^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $25.80$ | 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: | $3, 7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{668332} a^{8} - \frac{76995}{668332} a^{7} - \frac{105797}{668332} a^{6} - \frac{15401}{95476} a^{5} + \frac{37351}{95476} a^{4} + \frac{8925}{95476} a^{3} - \frac{12825}{95476} a^{2} + \frac{2399}{23869} a - \frac{5483}{23869}$
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: | \( 1296.65389295 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$He_3:C_2$ (as 9T11):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 54 |
| The 10 conjugacy class representatives for $C_3^2 : C_6$ |
| Character table for $C_3^2 : C_6$ |
Intermediate fields
| 3.1.567.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 9 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 sibling: | 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 | ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.9.16.5 | $x^{9} + 3 x^{8} + 6 x^{6} + 3$ | $9$ | $1$ | $16$ | $C_3^2 : C_6$ | $[2, 2]^{6}$ |
| $7$ | $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 7.2.1.2 | $x^{2} + 14$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.6.5.5 | $x^{6} + 56$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.7.2t1.1c1 | $1$ | $ 7 $ | $x^{2} - x + 2$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.7.3t1.1c1 | $1$ | $ 7 $ | $x^{3} - x^{2} - 2 x + 1$ | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.7.6t1.1c1 | $1$ | $ 7 $ | $x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.7.3t1.1c2 | $1$ | $ 7 $ | $x^{3} - x^{2} - 2 x + 1$ | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.7.6t1.1c2 | $1$ | $ 7 $ | $x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 2.3e4_7.3t2.1c1 | $2$ | $ 3^{4} \cdot 7 $ | $x^{3} - 3 x - 5$ | $S_3$ (as 3T2) | $1$ | $0$ |
| 2.3e4_7e2.6t5.3c1 | $2$ | $ 3^{4} \cdot 7^{2}$ | $x^{6} - 7 x^{3} + 63 x^{2} - 105 x + 56$ | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
| 2.3e4_7e2.6t5.3c2 | $2$ | $ 3^{4} \cdot 7^{2}$ | $x^{6} - 7 x^{3} + 63 x^{2} - 105 x + 56$ | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
| * | 6.3e12_7e5.9t13.1c1 | $6$ | $ 3^{12} \cdot 7^{5}$ | $x^{9} - 3 x^{8} + 3 x^{7} - 15 x^{6} + 21 x^{5} - 21 x^{4} + 21 x^{3} - 84 x^{2} + 84 x - 56$ | $C_3^2 : C_6$ (as 9T11) | $1$ | $0$ |
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 *.