Normalized defining polynomial
\( x^{9} - 105 x^{7} - 147 x^{6} + 3276 x^{5} + 6279 x^{4} - 35581 x^{3} - 63882 x^{2} + 131313 x + 164437 \)
Invariants
| Degree: | $9$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[9, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(87500953582971201=3^{12}\cdot 7^{8}\cdot 13^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $76.29$ | 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: | $3, 7, 13$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{13} a^{6} - \frac{1}{13} a^{4} - \frac{4}{13} a^{3}$, $\frac{1}{13} a^{7} - \frac{1}{13} a^{5} - \frac{4}{13} a^{4}$, $\frac{1}{923105301287} a^{8} + \frac{15723199649}{923105301287} a^{7} - \frac{4633881353}{923105301287} a^{6} + \frac{367922308892}{923105301287} a^{5} - \frac{276626582721}{923105301287} a^{4} - \frac{231220168663}{923105301287} a^{3} - \frac{13676767837}{71008100099} a^{2} - \frac{12460334254}{71008100099} a - \frac{188959213}{510849641}$
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
| Rank: | $8$ | 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: | \( 86246.0509855 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 27 |
| The 11 conjugacy class representatives for $C_9:C_3$ |
| Character table for $C_9:C_3$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | 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 | ${\href{/LocalNumberField/2.9.0.1}{9} }$ | R | ${\href{/LocalNumberField/5.9.0.1}{9} }$ | R | ${\href{/LocalNumberField/11.9.0.1}{9} }$ | R | ${\href{/LocalNumberField/17.9.0.1}{9} }$ | ${\href{/LocalNumberField/19.9.0.1}{9} }$ | ${\href{/LocalNumberField/23.9.0.1}{9} }$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/31.9.0.1}{9} }$ | ${\href{/LocalNumberField/37.9.0.1}{9} }$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }$ | ${\href{/LocalNumberField/53.9.0.1}{9} }$ | ${\href{/LocalNumberField/59.9.0.1}{9} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $3$ | 3.9.12.5 | $x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 27$ | $3$ | $3$ | $12$ | $C_9:C_3$ | $[2, 2]^{3}$ |
| $7$ | 7.9.8.2 | $x^{9} - 7$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ |
| $13$ | 13.3.0.1 | $x^{3} - 2 x + 6$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 13.3.2.3 | $x^{3} - 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.3.2.2 | $x^{3} - 13$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |