Normalized defining polynomial
\( x^{9} - 4x^{8} + 8x^{7} - 8x^{6} - 4x^{5} + 18x^{4} - 20x^{3} + 12x^{2} - 5x + 1 \)
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: | \(-1105805439\) \(\medspace = -\,3^{4}\cdot 239^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(10.11\) | 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: | $3^{4/3}239^{1/2}\approx 66.8899118207271$ | ||
Ramified primes: | \(3\), \(239\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-239}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $a^{8}-3a^{7}+5a^{6}-3a^{5}-7a^{4}+11a^{3}-9a^{2}+3a-2$, $a^{8}-4a^{7}+8a^{6}-8a^{5}-4a^{4}+18a^{3}-20a^{2}+12a-4$, $\frac{5}{3}a^{8}-6a^{7}+\frac{32}{3}a^{6}-8a^{5}-\frac{35}{3}a^{4}+\frac{80}{3}a^{3}-\frac{62}{3}a^{2}+\frac{22}{3}a-2$, $\frac{2}{3}a^{8}-\frac{8}{3}a^{7}+\frac{16}{3}a^{6}-5a^{5}-\frac{10}{3}a^{4}+\frac{38}{3}a^{3}-\frac{38}{3}a^{2}+\frac{16}{3}a-\frac{8}{3}$, $\frac{1}{3}a^{8}-\frac{4}{3}a^{7}+\frac{8}{3}a^{6}-3a^{5}-\frac{2}{3}a^{4}+\frac{16}{3}a^{3}-\frac{22}{3}a^{2}+\frac{20}{3}a-\frac{7}{3}$ | 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: | \( 10.1477527038 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 10.1477527038 \cdot 1}{2\cdot\sqrt{1105805439}}\cr\approx \mathstrut & 0.302782039955 \end{aligned}\]
Galois group
$C_3\wr S_3$ (as 9T20):
A solvable group of order 162 |
The 22 conjugacy class representatives for $C_3 \wr S_3 $ |
Character table for $C_3 \wr S_3 $ is not computed |
Intermediate fields
3.1.239.1 |
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 siblings: | 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 | ${\href{/padicField/2.9.0.1}{9} }$ | R | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.3.4.3 | $x^{3} + 6 x^{2} + 12$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(239\) | $\Q_{239}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{239}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{239}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $6$ | $2$ | $3$ | $3$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.239.2t1.a.a | $1$ | $ 239 $ | \(\Q(\sqrt{-239}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.2151.6t1.a.a | $1$ | $ 3^{2} \cdot 239 $ | 6.0.89570240559.4 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
1.2151.6t1.a.b | $1$ | $ 3^{2} \cdot 239 $ | 6.0.89570240559.4 | $C_6$ (as 6T1) | $0$ | $-1$ | |
* | 2.239.3t2.a.a | $2$ | $ 239 $ | 3.1.239.1 | $S_3$ (as 3T2) | $1$ | $0$ |
2.19359.6t5.a.a | $2$ | $ 3^{4} \cdot 239 $ | 6.0.89570240559.2 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
2.19359.6t5.a.b | $2$ | $ 3^{4} \cdot 239 $ | 6.0.89570240559.2 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
3.514089.18t86.b.a | $3$ | $ 3^{2} \cdot 239^{2}$ | 9.3.1105805439.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.41641209.18t86.b.a | $3$ | $ 3^{6} \cdot 239^{2}$ | 9.3.1105805439.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
* | 3.2151.9t20.b.a | $3$ | $ 3^{2} \cdot 239 $ | 9.3.1105805439.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ |
3.4626801.18t86.b.a | $3$ | $ 3^{4} \cdot 239^{2}$ | 9.3.1105805439.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.19359.9t20.b.a | $3$ | $ 3^{4} \cdot 239 $ | 9.3.1105805439.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ | |
3.19359.9t20.b.b | $3$ | $ 3^{4} \cdot 239 $ | 9.3.1105805439.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ | |
3.174231.9t20.b.a | $3$ | $ 3^{6} \cdot 239 $ | 9.3.1105805439.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ | |
3.4626801.18t86.b.b | $3$ | $ 3^{4} \cdot 239^{2}$ | 9.3.1105805439.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.41641209.18t86.b.b | $3$ | $ 3^{6} \cdot 239^{2}$ | 9.3.1105805439.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.514089.18t86.b.b | $3$ | $ 3^{2} \cdot 239^{2}$ | 9.3.1105805439.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.174231.9t20.b.b | $3$ | $ 3^{6} \cdot 239 $ | 9.3.1105805439.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ | |
* | 3.2151.9t20.b.b | $3$ | $ 3^{2} \cdot 239 $ | 9.3.1105805439.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ |
6.89570240559.9t11.a.a | $6$ | $ 3^{8} \cdot 239^{3}$ | 9.1.21407287493601.3 | $C_3^2 : C_6$ (as 9T11) | $1$ | $0$ |