Normalized defining polynomial
\( x^{9} - 735x^{6} - 225150x^{3} - 14706125 \)
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: | \(-619735169323054672183786016671875\) \(\medspace = -\,3^{19}\cdot 5^{6}\cdot 1801^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(4401.27\) | 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: | $3^{13/6}5^{2/3}1801^{2/3}\approx 4678.269070908049$ | ||
Ramified primes: | \(3\), \(5\), \(1801\) | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{5}a^{3}$, $\frac{1}{5}a^{4}$, $\frac{1}{15}a^{5}-\frac{1}{15}a^{4}+\frac{1}{15}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{75}a^{6}+\frac{1}{15}a^{3}+\frac{1}{3}$, $\frac{1}{3675}a^{7}+\frac{1}{15}a^{4}+\frac{10}{147}a$, $\frac{1}{180075}a^{8}-\frac{1}{245}a^{5}+\frac{1}{15}a^{4}-\frac{1}{15}a^{3}+\frac{2999}{7203}a^{2}-\frac{1}{3}a+\frac{1}{3}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{129402}$, which has order $24845184$ (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: | $\frac{2}{3675}a^{8}-\frac{2}{735}a^{7}+\frac{1}{75}a^{6}-\frac{7}{15}a^{5}+\frac{7}{3}a^{4}-\frac{173}{15}a^{3}-\frac{9388}{147}a^{2}+\frac{46940}{147}a-\frac{4730}{3}$, $\frac{1}{25}a^{6}-\frac{171}{5}a^{3}-4901$, $\frac{1}{5}a^{3}-24$, $\frac{104}{180075}a^{8}-\frac{4}{735}a^{7}+\frac{2}{75}a^{6}+\frac{361}{735}a^{5}+\frac{14}{3}a^{4}-\frac{343}{15}a^{3}+\frac{528454}{7203}a^{2}+\frac{96085}{147}a-\frac{9607}{3}$, $\frac{1}{60025}a^{8}+\frac{3}{245}a^{5}+\frac{11407}{2401}a^{2}-74$ (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: | \( 297351.2982801563 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 297351.2982801563 \cdot 24845184}{2\cdot\sqrt{619735169323054672183786016671875}}\cr\approx \mathstrut & 0.294447981541462 \end{aligned}\] (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
3.3.262731681.1, 3.1.19704876075.1 |
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: | 6.0.359121366466875.1 |
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 | R | ${\href{/padicField/7.3.0.1}{3} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.1.0.1}{1} }^{9}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.9.19.50 | $x^{9} + 6 x^{6} + 18 x^{2} + 21$ | $9$ | $1$ | $19$ | $S_3\times C_3$ | $[2, 5/2]_{2}$ |
\(5\) | 5.9.6.1 | $x^{9} + 9 x^{7} + 24 x^{6} + 27 x^{5} + 9 x^{4} - 186 x^{3} + 216 x^{2} - 504 x + 647$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
\(1801\) | Deg $3$ | $3$ | $1$ | $2$ | |||
Deg $3$ | $3$ | $1$ | $2$ | ||||
Deg $3$ | $3$ | $1$ | $2$ |