Normalized defining polynomial
\( x^{9} - 2 x^{8} + 941 x^{7} + 16825 x^{6} - 166443 x^{5} + 11732922 x^{4} + 165187300 x^{3} - 3066921559 x^{2} - 80668651817 x + 602452994221 \)
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: | \(-124988540775249857204408000=-\,2^{6}\cdot 5^{3}\cdot 7^{6}\cdot 13^{6}\cdot 31^{7}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $793.69$ | 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: | $2, 5, 7, 13, 31$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{15} a^{7} - \frac{1}{15} a^{6} - \frac{1}{5} a^{5} + \frac{1}{3} a^{4} - \frac{4}{15} a^{3} + \frac{1}{5} a^{2} + \frac{1}{3} a + \frac{7}{15}$, $\frac{1}{406747459280731473068957124548391682605} a^{8} - \frac{682185149504615161354409066315016418}{27116497285382098204597141636559445507} a^{7} + \frac{4752164906500442295128948018779262753}{406747459280731473068957124548391682605} a^{6} + \frac{64269529812681545216927037208259882561}{406747459280731473068957124548391682605} a^{5} + \frac{73717819637129165379005240429319643972}{406747459280731473068957124548391682605} a^{4} - \frac{54651264684993512073460109598551519728}{406747459280731473068957124548391682605} a^{3} - \frac{155074335079054162309403178969990888496}{406747459280731473068957124548391682605} a^{2} - \frac{13177527144538488040872511575918104146}{45194162142303497007661902727599075845} a - \frac{141769501084999437664456849966610791826}{406747459280731473068957124548391682605}$
Class group and class number
$C_{3}\times C_{3}\times C_{3}\times C_{21}\times C_{21}$, which has order $11907$ (assuming GRH)
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 (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 935147.1080403534 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3\times S_3$ (as 9T4):
| A solvable group of order 18 |
| The 9 conjugacy class representatives for $S_3\times C_3$ |
| Character table for $S_3\times C_3$ |
Intermediate fields
| 3.3.7958041.3, 3.1.620.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 6 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 | ${\href{/LocalNumberField/3.3.0.1}{3} }^{3}$ | R | R | ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ | R | ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ | R | ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.9.6.1 | $x^{9} - 4 x^{3} + 8$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
| $5$ | $\Q_{5}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{5}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{5}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $7$ | 7.3.2.2 | $x^{3} - 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 7.6.4.3 | $x^{6} + 56 x^{3} + 1323$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| $13$ | 13.3.2.3 | $x^{3} - 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 13.3.2.3 | $x^{3} - 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.3.2.3 | $x^{3} - 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| $31$ | 31.3.2.3 | $x^{3} - 1519$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 31.6.5.6 | $x^{6} + 521017$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |