magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![112, -400, 560, -336, 72, 16, -12, 4, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 4*x^7 - 12*x^6 + 16*x^5 + 72*x^4 - 336*x^3 + 560*x^2 - 400*x + 112)
gp: K = bnfinit(x^9 - 3*x^8 + 4*x^7 - 12*x^6 + 16*x^5 + 72*x^4 - 336*x^3 + 560*x^2 - 400*x + 112, 1)
Normalized defining polynomial
\( x^{9} - 3 x^{8} + 4 x^{7} - 12 x^{6} + 16 x^{5} + 72 x^{4} - 336 x^{3} + 560 x^{2} - 400 x + 112 \)
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: | \(47469233803264=2^{12}\cdot 7^{4}\cdot 13^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $33.08$ | 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: | $2, 7, 13$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2128} a^{8} + \frac{15}{2128} a^{7} + \frac{1}{266} a^{6} - \frac{67}{1064} a^{5} + \frac{33}{266} a^{4} + \frac{9}{532} a^{3} - \frac{47}{133} a^{2} + \frac{107}{266} a + \frac{1}{19}$
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: | \( 9268.19273402 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$\mathrm{P}\Gamma\mathrm{L}(2,8)$ (as 9T32):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A non-solvable group of order 1512 |
| The 11 conjugacy class representatives for $\mathrm{P}\Gamma\mathrm{L}(2,8)$ |
| Character table for $\mathrm{P}\Gamma\mathrm{L}(2,8)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 27 sibling: | data not computed |
| Degree 28 sibling: | data not computed |
| 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 | ${\href{/LocalNumberField/3.9.0.1}{9} }$ | ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/11.9.0.1}{9} }$ | R | ${\href{/LocalNumberField/17.9.0.1}{9} }$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.9.0.1}{9} }$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/37.9.0.1}{9} }$ | ${\href{/LocalNumberField/41.9.0.1}{9} }$ | ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }$ | ${\href{/LocalNumberField/53.9.0.1}{9} }$ | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.8.12.28 | $x^{8} + 2 x^{5} + 2$ | $8$ | $1$ | $12$ | $C_2^3:(C_7: C_3)$ | $[12/7, 12/7, 12/7]_{7}^{3}$ | |
| $7$ | $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 7.3.2.2 | $x^{3} - 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.3.2.2 | $x^{3} - 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| $13$ | 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.7.6.1 | $x^{7} - 13$ | $7$ | $1$ | $6$ | $D_{7}$ | $[\ ]_{7}^{2}$ |