magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 15, -3, -39, 48, -18, 3, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 3*x^7 - 18*x^6 + 48*x^5 - 39*x^4 - 3*x^3 + 15*x^2 - 6*x + 1)
gp: K = bnfinit(x^9 - 3*x^8 + 3*x^7 - 18*x^6 + 48*x^5 - 39*x^4 - 3*x^3 + 15*x^2 - 6*x + 1, 1)
Normalized defining polynomial
\( x^{9} - 3 x^{8} + 3 x^{7} - 18 x^{6} + 48 x^{5} - 39 x^{4} - 3 x^{3} + 15 x^{2} - 6 x + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $9$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[3, 3]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-45370442231868096=-\,2^{6}\cdot 3^{16}\cdot 7^{4}\cdot 19^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $70.92$ | 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: | $2, 3, 7, 19$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{6}\times C_{18}$, which has order $108$
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $5$ | 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: | \( a \), \( a - 1 \), \( a^{8} - 5 a^{7} + 9 a^{6} - 25 a^{5} + 87 a^{4} - 138 a^{3} + 94 a^{2} - 28 a + 4 \), \( a^{8} - 2 a^{7} + a^{6} - 17 a^{5} + 31 a^{4} - 8 a^{3} - 12 a^{2} + 6 a - 1 \), \( a^{6} - a^{5} - 17 a^{3} + 14 a^{2} + 6 a - 5 \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 1111.19177718 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3^2:S_3$ (as 9T12):
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 $(C_3^2:C_3):C_2$ |
| Character table for $(C_3^2:C_3):C_2$ |
Intermediate fields
| 3.1.6156.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 9 siblings: | 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 | R | R | ${\href{/LocalNumberField/5.3.0.1}{3} }^{3}$ | R | ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ | R | ${\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.9.6.1 | $x^{9} - 4 x^{3} + 8$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
| $3$ | 3.9.16.14 | $x^{9} + 3 x^{8} + 6$ | $9$ | $1$ | $16$ | $S_3\times C_3$ | $[2, 2]^{2}$ |
| $7$ | 7.3.0.1 | $x^{3} - x + 2$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 7.3.2.1 | $x^{3} + 14$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.3.2.3 | $x^{3} - 28$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| $19$ | 19.3.0.1 | $x^{3} - x + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 19.6.3.2 | $x^{6} - 361 x^{2} + 27436$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |