Normalized defining polynomial
\( x^{9} - 507x^{6} - 3429205026x^{3} - 4826809 \)
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: |
\(-535650742595126498202296828232161427\)
\(\medspace = -\,3^{15}\cdot 13^{6}\cdot 4447^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(9329.87\) | 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: | $3^{31/18}13^{2/3}4447^{2/3}\approx 9917.049525086115$ | ||
Ramified primes: |
\(3\), \(13\), \(4447\)
| 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 \(a\))
$1$, $a$, $\frac{1}{13}a^{2}$, $\frac{1}{507}a^{3}-\frac{1}{3}$, $\frac{1}{507}a^{4}-\frac{1}{3}a$, $\frac{1}{19773}a^{5}+\frac{1}{1521}a^{4}+\frac{1}{1521}a^{3}-\frac{4}{117}a^{2}-\frac{4}{9}a-\frac{4}{9}$, $\frac{1}{19792773}a^{6}+\frac{16}{16731}a^{3}+\frac{229}{693}$, $\frac{1}{59378319}a^{7}-\frac{1}{59378319}a^{6}+\frac{49}{50193}a^{4}-\frac{49}{50193}a^{3}+\frac{691}{2079}a-\frac{691}{2079}$, $\frac{1}{771918147}a^{8}-\frac{1}{59378319}a^{6}+\frac{16}{652509}a^{5}-\frac{1}{1521}a^{4}+\frac{17}{50193}a^{3}-\frac{464}{27027}a^{2}+\frac{4}{9}a-\frac{460}{2079}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{12}\times C_{36}\times C_{1510848}$, which has order $652686336$ (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: |
$\frac{1384}{257306049}a^{8}+\frac{2}{6597591}a^{7}-\frac{1}{19792773}a^{6}-\frac{593}{217503}a^{5}-\frac{1}{5577}a^{4}+\frac{17}{16731}a^{3}-\frac{12782438}{693}a^{2}-\frac{239782}{231}a+\frac{79235}{693}$, $\frac{2}{19792773}a^{6}-\frac{1}{16731}a^{3}-\frac{239782}{693}$, $\frac{1}{19792773}a^{6}-\frac{50}{16731}a^{3}-\frac{2}{693}$, $\frac{1388}{257306049}a^{8}+\frac{2}{6597591}a^{7}-\frac{1}{19792773}a^{6}-\frac{595}{217503}a^{5}-\frac{1}{5577}a^{4}-\frac{16}{16731}a^{3}-\frac{166651951}{9009}a^{2}-\frac{240475}{231}a+\frac{80159}{693}$, $\frac{1}{371293}a^{8}+\frac{3}{2197}a^{5}+\frac{120067}{13}a^{2}-116$
| 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: | \( 447510.71870329685 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 447510.71870329685 \cdot 652686336}{2\cdot\sqrt{535650742595126498202296828232161427}}\cr\approx \mathstrut & 0.395974029894405 \end{aligned}\] (assuming GRH)
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
3.3.1601840529.1, 3.1.90237016467.1 |
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 |
Minimal sibling: | 6.0.11117290665750867.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.3.0.1}{3} }^{3}$ | ${\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.1.0.1}{1} }^{9}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\)
| 3.9.15.28 | $x^{9} + 6 x^{8} + 6 x^{7} + 3 x^{3} + 21$ | $9$ | $1$ | $15$ | $S_3\times C_3$ | $[3/2, 2]_{2}$ |
\(13\)
| 13.9.6.1 | $x^{9} + 6 x^{7} + 72 x^{6} + 12 x^{5} + 54 x^{4} - 2125 x^{3} + 288 x^{2} - 2160 x + 13928$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
\(4447\)
| Deg $3$ | $3$ | $1$ | $2$ | |||
Deg $3$ | $3$ | $1$ | $2$ | ||||
Deg $3$ | $3$ | $1$ | $2$ |