Normalized defining polynomial
\( x^{9} - 3x^{6} + 3x^{5} - 3x^{4} - 6x^{2} + 6x - 2 \)
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: | \(13060694016\) \(\medspace = 2^{13}\cdot 3^{13}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.30\) | 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: | $2^{11/4}3^{85/54}\approx 37.91898338618923$ | ||
Ramified primes: | \(2\), \(3\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{6}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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: | $a^{8}+a^{7}-3a^{5}+a^{4}-a^{3}-2a^{2}-7a+3$, $\frac{1}{2}a^{8}+\frac{1}{2}a^{7}+\frac{1}{2}a^{6}-2a^{5}-\frac{1}{2}a^{4}-a^{3}+a^{2}-3a$, $a^{8}+a^{6}-3a^{5}+2a^{4}-4a^{3}+a^{2}-6a+1$, $\frac{1}{2}a^{8}+\frac{1}{2}a^{7}+\frac{1}{2}a^{6}-a^{5}-\frac{1}{2}a^{4}-a^{3}-a^{2}-2a-2$ | 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: | \( 91.4873489599 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 91.4873489599 \cdot 1}{2\cdot\sqrt{13060694016}}\cr\approx \mathstrut & 1.24766326727 \end{aligned}\]
Galois group
$S_3\wr S_3$ (as 9T31):
A solvable group of order 1296 |
The 22 conjugacy class representatives for $S_3\wr S_3$ |
Character table for $S_3\wr S_3$ is not computed |
Intermediate fields
3.1.216.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 27 siblings: | data not computed |
Degree 36 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 | R | R | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
2.4.10.4 | $x^{4} + 4 x^{3} + 8 x^{2} + 10$ | $4$ | $1$ | $10$ | $D_{4}$ | $[2, 3, 7/2]$ | |
\(3\) | 3.9.13.7 | $x^{9} + 3 x^{5} + 6 x^{3} + 3$ | $9$ | $1$ | $13$ | $C_3^2 : D_{6} $ | $[3/2, 3/2, 5/3]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.24.2t1.b.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{-6}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.24.2t1.a.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{6}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.864.6t3.a.a | $2$ | $ 2^{5} \cdot 3^{3}$ | 6.2.4478976.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.216.3t2.b.a | $2$ | $ 2^{3} \cdot 3^{3}$ | 3.1.216.1 | $S_3$ (as 3T2) | $1$ | $0$ |
3.13824.4t5.c.a | $3$ | $ 2^{9} \cdot 3^{3}$ | 4.2.13824.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.82944.6t11.a.a | $3$ | $ 2^{10} \cdot 3^{4}$ | 6.0.746496.1 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
3.82944.6t8.h.a | $3$ | $ 2^{10} \cdot 3^{4}$ | 4.2.13824.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.3456.6t11.a.a | $3$ | $ 2^{7} \cdot 3^{3}$ | 6.0.746496.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
* | 6.60466176.9t31.a.a | $6$ | $ 2^{10} \cdot 3^{10}$ | 9.1.13060694016.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |
6.15479341056.18t300.a.a | $6$ | $ 2^{18} \cdot 3^{10}$ | 9.1.13060694016.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.3869835264.18t319.a.a | $6$ | $ 2^{16} \cdot 3^{10}$ | 9.1.13060694016.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.3869835264.18t311.a.a | $6$ | $ 2^{16} \cdot 3^{10}$ | 9.1.13060694016.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.356...024.24t2893.c.a | $8$ | $ 2^{26} \cdot 3^{12}$ | 9.1.13060694016.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.15479341056.12t213.a.a | $8$ | $ 2^{18} \cdot 3^{10}$ | 9.1.13060694016.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.159...424.36t2217.a.a | $12$ | $ 2^{37} \cdot 3^{19}$ | 9.1.13060694016.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
12.998...464.36t2214.a.a | $12$ | $ 2^{33} \cdot 3^{19}$ | 9.1.13060694016.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.239...136.36t2210.a.a | $12$ | $ 2^{36} \cdot 3^{20}$ | 9.1.13060694016.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.998...464.36t2216.a.a | $12$ | $ 2^{33} \cdot 3^{19}$ | 9.1.13060694016.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.623...904.18t315.a.a | $12$ | $ 2^{29} \cdot 3^{19}$ | 9.1.13060694016.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
16.447...864.24t2912.a.a | $16$ | $ 2^{44} \cdot 3^{26}$ | 9.1.13060694016.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |