Normalized defining polynomial
\( x^{9} - 9x^{7} - 25x^{6} + 159x^{5} + 216x^{4} - 794x^{3} - 3525x^{2} - 75x + 14299 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-3789758442045443136192\) \(\medspace = -\,2^{6}\cdot 3^{9}\cdot 7^{4}\cdot 11^{6}\cdot 29^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(249.82\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $2^{2/3}3^{7/6}7^{2/3}11^{2/3}29^{2/3}\approx 977.0585953813884$ | ||
Ramified primes: | \(2\), \(3\), \(7\), \(11\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6593555569}a^{8}-\frac{2915399129}{6593555569}a^{7}+\frac{1904022818}{6593555569}a^{6}+\frac{621291301}{6593555569}a^{5}-\frac{1576471654}{6593555569}a^{4}-\frac{568228571}{6593555569}a^{3}+\frac{700211182}{6593555569}a^{2}-\frac{1536402293}{6593555569}a+\frac{1534091225}{6593555569}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}\times C_{387}$, which has order $1161$ (assuming GRH)
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{694843}{6593555569}a^{8}-\frac{5972308}{6593555569}a^{7}+\frac{2007724}{6593555569}a^{6}+\frac{47691606}{6593555569}a^{5}+\frac{280308786}{6593555569}a^{4}-\frac{943932064}{6593555569}a^{3}-\frac{1627102084}{6593555569}a^{2}-\frac{2389853778}{6593555569}a+\frac{18575101428}{6593555569}$, $\frac{3886612}{6593555569}a^{8}+\frac{5765552}{6593555569}a^{7}-\frac{37487706}{6593555569}a^{6}-\frac{68739244}{6593555569}a^{5}+\frac{393508261}{6593555569}a^{4}+\frac{1080822822}{6593555569}a^{3}-\frac{1317275952}{6593555569}a^{2}-\frac{11923292156}{6593555569}a-\frac{16252866758}{6593555569}$, $\frac{32592167}{6593555569}a^{8}+\frac{162168661}{6593555569}a^{7}-\frac{1021356675}{6593555569}a^{6}-\frac{1302116149}{6593555569}a^{5}+\frac{10405499008}{6593555569}a^{4}+\frac{26750225756}{6593555569}a^{3}-\frac{129040189009}{6593555569}a^{2}-\frac{133337646639}{6593555569}a+\frac{471673596558}{6593555569}$, $\frac{5477323}{6593555569}a^{8}+\frac{70664276}{6593555569}a^{7}+\frac{252735742}{6593555569}a^{6}-\frac{19160505}{6593555569}a^{5}-\frac{3198806670}{6593555569}a^{4}-\frac{8792404794}{6593555569}a^{3}-\frac{3842905582}{6593555569}a^{2}+\frac{29510902044}{6593555569}a+\frac{50772546300}{6593555569}$, $\frac{37448270}{6593555569}a^{8}-\frac{176831465}{6593555569}a^{7}+\frac{109904017}{6593555569}a^{6}+\frac{465503110}{6593555569}a^{5}+\frac{2118926648}{6593555569}a^{4}-\frac{6054466678}{6593555569}a^{3}-\frac{22528689459}{6593555569}a^{2}+\frac{41521100357}{6593555569}a+\frac{12739290529}{6593555569}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 12207.216264322764 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 12207.216264322764 \cdot 1161}{2\cdot\sqrt{3789758442045443136192}}\cr\approx \mathstrut & 0.228424503840834 \end{aligned}\] (assuming GRH)
Galois group
$C_3^2:S_3$ (as 9T12):
A solvable group of order 54 |
The 10 conjugacy class representatives for $(C_3^2:C_3):C_2$ |
Character table for $(C_3^2:C_3):C_2$ |
Intermediate fields
3.1.13068.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 9 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 27 sibling: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
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 | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | R | ${\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | R | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
\(3\) | 3.9.9.6 | $x^{9} - 6 x^{8} + 45 x^{7} + 594 x^{6} + 99 x^{5} + 108 x^{4} - 54 x^{3} + 27 x^{2} + 81 x + 27$ | $3$ | $3$ | $9$ | $S_3\times C_3$ | $[3/2]_{2}^{3}$ |
\(7\) | 7.3.2.1 | $x^{3} + 14$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(11\) | 11.9.6.1 | $x^{9} + 6 x^{7} + 60 x^{6} + 12 x^{5} + 42 x^{4} - 1465 x^{3} + 240 x^{2} - 1560 x + 8088$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
\(29\) | 29.3.0.1 | $x^{3} + 2 x + 27$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
29.6.4.2 | $x^{6} - 1914 x^{3} - 2069701$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |