Normalized defining polynomial
\( x^{9} + 9x^{7} - 5x^{6} + 27x^{5} - 30x^{4} + 31x^{3} - 45x^{2} + 12x + 9 \)
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: | \(134573648589\) \(\medspace = 3^{8}\cdot 29^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(17.24\) | 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: | $3^{7/6}29^{3/4}\approx 45.02357354987992$ | ||
Ramified primes: | \(3\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{29}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{4}$, $\frac{1}{9}a^{8}+\frac{1}{9}a^{7}+\frac{1}{9}a^{6}-\frac{1}{9}a^{5}-\frac{1}{9}a^{4}-\frac{4}{9}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$
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: | $\frac{1}{3}a^{6}+2a^{4}-\frac{1}{3}a^{3}+3a^{2}-a-1$, $\frac{2}{9}a^{8}+\frac{2}{9}a^{7}+\frac{17}{9}a^{6}+\frac{7}{9}a^{5}+\frac{43}{9}a^{4}-\frac{5}{9}a^{3}+\frac{10}{3}a^{2}-\frac{8}{3}a-1$, $\frac{2}{9}a^{8}+\frac{2}{9}a^{7}+\frac{20}{9}a^{6}+\frac{13}{9}a^{5}+\frac{61}{9}a^{4}+\frac{19}{9}a^{3}+\frac{14}{3}a^{2}-\frac{2}{3}a-6$, $\frac{1}{3}a^{8}+\frac{10}{3}a^{6}-a^{5}+11a^{4}-\frac{19}{3}a^{3}+\frac{35}{3}a^{2}-10a-2$ | 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: | \( 177.351346168 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 177.351346168 \cdot 1}{2\cdot\sqrt{134573648589}}\cr\approx \mathstrut & 0.753483851587 \end{aligned}\]
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
3.1.87.1 |
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 | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | R | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | R | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.6.7.3 | $x^{6} + 3 x^{3} + 6 x^{2} + 6$ | $6$ | $1$ | $7$ | $S_3\times C_3$ | $[3/2]_{2}^{3}$ | |
\(29\) | $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
29.4.3.3 | $x^{4} + 58$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.87.2t1.a.a | $1$ | $ 3 \cdot 29 $ | \(\Q(\sqrt{-87}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.29.2t1.a.a | $1$ | $ 29 $ | \(\Q(\sqrt{29}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.87.6t3.a.a | $2$ | $ 3 \cdot 29 $ | 6.2.219501.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.87.3t2.a.a | $2$ | $ 3 \cdot 29 $ | 3.1.87.1 | $S_3$ (as 3T2) | $1$ | $0$ |
3.73167.4t5.a.a | $3$ | $ 3 \cdot 29^{3}$ | 4.2.73167.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.2523.6t11.b.a | $3$ | $ 3 \cdot 29^{2}$ | 6.0.19096587.1 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
3.7569.6t8.a.a | $3$ | $ 3^{2} \cdot 29^{2}$ | 4.2.73167.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.219501.6t11.b.a | $3$ | $ 3^{2} \cdot 29^{3}$ | 6.0.19096587.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
* | 6.1546823547.9t31.a.a | $6$ | $ 3^{7} \cdot 29^{4}$ | 9.1.134573648589.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |
6.1546823547.18t300.a.a | $6$ | $ 3^{7} \cdot 29^{4}$ | 9.1.134573648589.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.1546823547.18t319.a.a | $6$ | $ 3^{7} \cdot 29^{4}$ | 9.1.134573648589.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.1546823547.18t311.a.a | $6$ | $ 3^{7} \cdot 29^{4}$ | 9.1.134573648589.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.390...081.24t2893.a.a | $8$ | $ 3^{8} \cdot 29^{6}$ | 9.1.134573648589.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.390...081.12t213.a.a | $8$ | $ 3^{8} \cdot 29^{6}$ | 9.1.134573648589.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.208...183.36t2217.a.a | $12$ | $ 3^{15} \cdot 29^{9}$ | 9.1.134573648589.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
12.187...647.36t2214.a.a | $12$ | $ 3^{17} \cdot 29^{9}$ | 9.1.134573648589.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.201...769.36t2210.a.a | $12$ | $ 3^{14} \cdot 29^{10}$ | 9.1.134573648589.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.187...647.36t2216.a.a | $12$ | $ 3^{17} \cdot 29^{9}$ | 9.1.134573648589.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.208...183.18t315.a.a | $12$ | $ 3^{15} \cdot 29^{9}$ | 9.1.134573648589.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
16.152...561.24t2912.a.a | $16$ | $ 3^{16} \cdot 29^{12}$ | 9.1.134573648589.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |