magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-48, 144, -144, 48, 0, 0, 0, 0, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 48*x^3 - 144*x^2 + 144*x - 48)
gp: K = bnfinit(x^9 - 3*x^8 + 48*x^3 - 144*x^2 + 144*x - 48, 1)
Normalized defining polynomial
\( x^{9} - 3 x^{8} + 48 x^{3} - 144 x^{2} + 144 x - 48 \)
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: | \(119191893590016=2^{14}\cdot 3^{16}\cdot 13^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $36.65$ | 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: | $2, 3, 13$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a$, $\frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{16} a^{8} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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 (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 25150.0469199 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$A_9$ (as 9T33):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A non-solvable group of order 181440 |
| The 18 conjugacy class representatives for $A_9$ |
| Character table for $A_9$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 36 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 | ${\href{/LocalNumberField/5.9.0.1}{9} }$ | ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | ${\href{/LocalNumberField/11.7.0.1}{7} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }$ | ${\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.9.0.1}{9} }$ | ${\href{/LocalNumberField/43.9.0.1}{9} }$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | ${\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.7.0.1}{7} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.2.3.2 | $x^{2} + 6$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.6.11.8 | $x^{6} + 4 x^{4} + 2 x^{2} + 6$ | $6$ | $1$ | $11$ | $S_4\times C_2$ | $[8/3, 8/3, 3]_{3}^{2}$ | |
| $3$ | 3.9.16.7 | $x^{9} + 3 x^{8} + 3 x^{3} + 3$ | $9$ | $1$ | $16$ | $(C_9:C_3):C_2$ | $[3/2, 2, 13/6]_{2}$ |
| $13$ | 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.3.2.3 | $x^{3} - 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.4.0.1 | $x^{4} + x^{2} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |