magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -27, 3, 38, -60, 27, -5, 3, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 3*x^7 - 5*x^6 + 27*x^5 - 60*x^4 + 38*x^3 + 3*x^2 - 27*x + 8)
gp: K = bnfinit(x^9 - 3*x^8 + 3*x^7 - 5*x^6 + 27*x^5 - 60*x^4 + 38*x^3 + 3*x^2 - 27*x + 8, 1)
Normalized defining polynomial
\( x^{9} - 3 x^{8} + 3 x^{7} - 5 x^{6} + 27 x^{5} - 60 x^{4} + 38 x^{3} + 3 x^{2} - 27 x + 8 \)
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: | \(-409819132827=-\,3^{15}\cdot 13^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $19.51$ | 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 not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{39} a^{6} + \frac{6}{13} a^{5} + \frac{2}{13} a^{3} + \frac{3}{13} a^{2} - \frac{4}{13} a + \frac{14}{39}$, $\frac{1}{39} a^{7} - \frac{4}{13} a^{5} + \frac{2}{13} a^{4} + \frac{6}{13} a^{3} - \frac{6}{13} a^{2} - \frac{4}{39} a - \frac{6}{13}$, $\frac{1}{117} a^{8} - \frac{1}{117} a^{7} + \frac{1}{117} a^{6} + \frac{19}{39} a^{5} + \frac{17}{39} a^{4} + \frac{14}{39} a^{3} + \frac{14}{117} a^{2} - \frac{14}{117} a + \frac{44}{117}$
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: | $5$ | 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: | 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: | \( 768.465748793 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$AGL(2,3)$ (as 9T26):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 432 |
| The 11 conjugacy class representatives for $((C_3^2:Q_8):C_3):C_2$ |
| Character table for $((C_3^2:Q_8):C_3):C_2$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 12 sibling: | data not computed |
| Degree 18 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 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 | ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ | ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.3.4.4 | $x^{3} + 3 x^{2} + 3$ | $3$ | $1$ | $4$ | $S_3$ | $[2]^{2}$ |
| 3.6.11.6 | $x^{6} + 6 x^{3} + 15$ | $6$ | $1$ | $11$ | $S_3^2$ | $[2, 5/2]_{2}^{2}$ | |
| $13$ | 13.3.0.1 | $x^{3} - 2 x + 6$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 13.3.2.2 | $x^{3} - 13$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.3.2.3 | $x^{3} - 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |