Normalized defining polynomial
\( x^{9} - 3x^{8} - 9x^{7} + 15x^{6} + 150x^{5} - 450x^{4} + 300x^{3} + 150x^{2} - 100 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(49601160000000\) \(\medspace = 2^{9}\cdot 3^{11}\cdot 5^{7}\cdot 7\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(33.24\) | 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: | $2^{25/12}3^{37/18}5^{13/14}7^{1/2}\approx 478.0657883792325$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{210}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{2}$, $\frac{1}{45}a^{6}-\frac{1}{15}a^{5}+\frac{2}{15}a^{4}-\frac{1}{9}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{9}$, $\frac{1}{90}a^{7}-\frac{1}{90}a^{6}-\frac{1}{6}a^{5}+\frac{7}{90}a^{4}-\frac{1}{9}a^{3}-\frac{1}{3}a^{2}+\frac{1}{9}a+\frac{2}{9}$, $\frac{1}{90}a^{8}+\frac{2}{45}a^{5}+\frac{1}{30}a^{4}+\frac{4}{9}a^{2}-\frac{1}{3}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | 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: | $\frac{1}{90}a^{8}-\frac{1}{30}a^{7}-\frac{7}{90}a^{6}+\frac{19}{90}a^{5}+\frac{22}{15}a^{4}-\frac{52}{9}a^{3}+\frac{43}{9}a^{2}+\frac{11}{3}a-\frac{31}{9}$, $\frac{1}{90}a^{8}+\frac{1}{90}a^{7}-\frac{1}{18}a^{6}-\frac{29}{90}a^{5}+\frac{23}{45}a^{4}-\frac{2}{9}a^{3}+\frac{46}{9}a^{2}-\frac{92}{9}a-\frac{11}{9}$, $\frac{109}{45}a^{8}-\frac{206}{45}a^{7}-\frac{235}{9}a^{6}+\frac{289}{45}a^{5}+\frac{16159}{45}a^{4}-\frac{6304}{9}a^{3}+\frac{707}{9}a^{2}+\frac{3599}{9}a+\frac{2471}{9}$, $\frac{1611}{10}a^{8}-\frac{5981}{90}a^{7}-\frac{148963}{90}a^{6}-\frac{18733}{10}a^{5}+\frac{884483}{45}a^{4}-\frac{189370}{9}a^{3}-9803a^{2}+\frac{7666}{9}a+\frac{49829}{9}$ | 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: | \( 20551.4087704 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 20551.4087704 \cdot 1}{2\cdot\sqrt{49601160000000}}\cr\approx \mathstrut & 4.54794448011 \end{aligned}\]
Galois group
A non-solvable group of order 362880 |
The 30 conjugacy class representatives for $S_9$ |
Character table for $S_9$ is not computed |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 18 sibling: | data not computed |
Degree 36 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 | R | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.9.0.1}{9} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.9.0.1}{9} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.1 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.4.4.5 | $x^{4} + 2 x + 2$ | $4$ | $1$ | $4$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
\(3\) | 3.3.5.2 | $x^{3} + 18 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ |
3.6.6.3 | $x^{6} + 18 x^{5} + 120 x^{4} + 386 x^{3} + 723 x^{2} + 732 x + 305$ | $3$ | $2$ | $6$ | $D_{6}$ | $[3/2]_{2}^{2}$ | |
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.7.6.1 | $x^{7} + 5$ | $7$ | $1$ | $6$ | $F_7$ | $[\ ]_{7}^{6}$ | |
\(7\) | 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.7.0.1 | $x^{7} + 6 x + 4$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |