Normalized defining polynomial
\( x^{9} - 48x^{6} - 1041x^{3} - 4096 \)
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: | \(-1297978913113439283\) \(\medspace = -\,3^{15}\cdot 67^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(102.94\) | 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))
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Galois root discriminant: | $3^{31/18}67^{2/3}\approx 109.41882061656858$ | ||
Ramified primes: | \(3\), \(67\) | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{6}a^{4}-\frac{1}{6}a$, $\frac{1}{18}a^{5}+\frac{1}{18}a^{4}-\frac{1}{9}a^{3}-\frac{1}{18}a^{2}-\frac{1}{18}a+\frac{1}{9}$, $\frac{1}{18}a^{6}+\frac{1}{18}a^{3}-\frac{1}{9}$, $\frac{1}{432}a^{7}+\frac{1}{54}a^{6}+\frac{1}{27}a^{4}+\frac{7}{54}a^{3}-\frac{17}{432}a-\frac{4}{27}$, $\frac{1}{6912}a^{8}+\frac{1}{54}a^{6}-\frac{11}{432}a^{5}+\frac{1}{18}a^{4}+\frac{1}{54}a^{3}+\frac{1903}{6912}a^{2}-\frac{1}{18}a-\frac{1}{27}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{126}$, which has order $126$ (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: | $\frac{11}{6912}a^{8}-\frac{1}{108}a^{7}+\frac{1}{54}a^{6}-\frac{1}{432}a^{5}-\frac{5}{54}a^{4}+\frac{1}{54}a^{3}-\frac{1723}{6912}a^{2}+\frac{11}{108}a-\frac{28}{27}$, $\frac{5}{6912}a^{8}-\frac{1}{108}a^{7}+\frac{1}{54}a^{6}+\frac{17}{432}a^{5}-\frac{11}{54}a^{4}+\frac{31}{54}a^{3}+\frac{1451}{6912}a^{2}-\frac{85}{108}a+\frac{92}{27}$, $\frac{1}{108}a^{8}+\frac{1}{54}a^{7}-\frac{14}{27}a^{5}-\frac{28}{27}a^{4}-\frac{593}{108}a^{2}-\frac{593}{54}a-1$, $\frac{7}{3456}a^{8}+\frac{1}{108}a^{7}-\frac{29}{216}a^{5}-\frac{14}{27}a^{4}-\frac{1271}{3456}a^{2}-\frac{377}{108}a+1$, $\frac{25}{3456}a^{8}+\frac{1}{108}a^{7}-\frac{83}{216}a^{5}-\frac{14}{27}a^{4}-\frac{17705}{3456}a^{2}-\frac{809}{108}a+1$ (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: | \( 44046.7139727 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 44046.7139727 \cdot 126}{2\cdot\sqrt{1297978913113439283}}\cr\approx \mathstrut & 4.83336771910 \end{aligned}\] (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
3.3.363609.2, 3.1.121203.1 |
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: | 6.0.88356987.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{3}{,}\,{\href{/padicField/2.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.3.0.1}{3} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.9.15.27 | $x^{9} + 6 x^{8} + 6 x^{7} + 3 x^{3} + 3$ | $9$ | $1$ | $15$ | $S_3\times C_3$ | $[3/2, 2]_{2}$ |
\(67\) | 67.9.6.1 | $x^{9} + 18 x^{7} + 396 x^{6} + 108 x^{5} + 1134 x^{4} - 65097 x^{3} + 14256 x^{2} - 163944 x + 2314440$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |