Normalized defining polynomial
\( x^{9} + 6x^{5} - 6x^{2} + 2 \)
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: |
\(5578004736\)
\(\medspace = 2^{8}\cdot 3^{12}\cdot 41\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.10\) | 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: | $2^{8/9}3^{25/18}41^{1/2}\approx 54.5308862563819$ | ||
Ramified primes: |
\(2\), \(3\), \(41\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{41}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{89}a^{8}-\frac{27}{89}a^{7}+\frac{17}{89}a^{6}-\frac{14}{89}a^{5}+\frac{28}{89}a^{4}-\frac{44}{89}a^{3}+\frac{31}{89}a^{2}-\frac{42}{89}a-\frac{23}{89}$
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: |
$\frac{49}{89}a^{8}+\frac{12}{89}a^{7}+\frac{32}{89}a^{6}+\frac{26}{89}a^{5}+\frac{304}{89}a^{4}+\frac{69}{89}a^{3}+\frac{184}{89}a^{2}-\frac{189}{89}a-\frac{59}{89}$, $\frac{36}{89}a^{8}+\frac{7}{89}a^{7}-\frac{11}{89}a^{6}+\frac{30}{89}a^{5}+\frac{207}{89}a^{4}+\frac{18}{89}a^{3}-\frac{41}{89}a^{2}-\frac{88}{89}a-\frac{27}{89}$, $\frac{21}{89}a^{8}-\frac{33}{89}a^{7}+\frac{1}{89}a^{6}-\frac{27}{89}a^{5}+\frac{143}{89}a^{4}-\frac{212}{89}a^{3}+\frac{28}{89}a^{2}-\frac{259}{89}a+\frac{229}{89}$, $\frac{130}{89}a^{8}+\frac{50}{89}a^{7}+\frac{74}{89}a^{6}+\frac{49}{89}a^{5}+\frac{792}{89}a^{4}+\frac{332}{89}a^{3}+\frac{470}{89}a^{2}-\frac{476}{89}a-\frac{231}{89}$
| 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: | \( 32.7573550578 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 32.7573550578 \cdot 1}{2\cdot\sqrt{5578004736}}\cr\approx \mathstrut & 0.683579235666 \end{aligned}\]
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 $S_3\wr S_3$ |
Intermediate fields
3.1.108.1 |
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 | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.9.0.1}{9} }$ | R | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| 2.9.8.1 | $x^{9} + 2$ | $9$ | $1$ | $8$ | $(C_9:C_3):C_2$ | $[\ ]_{9}^{6}$ |
\(3\)
| 3.9.12.24 | $x^{9} + 3 x^{4} + 3$ | $9$ | $1$ | $12$ | $S_3^2$ | $[3/2, 3/2]_{2}^{2}$ |
\(41\)
| $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
41.2.1.1 | $x^{2} + 41$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
41.4.0.1 | $x^{4} + 23 x + 6$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.123.2t1.a.a | $1$ | $ 3 \cdot 41 $ | \(\Q(\sqrt{-123}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.41.2t1.a.a | $1$ | $ 41 $ | \(\Q(\sqrt{41}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.181548.6t3.b.a | $2$ | $ 2^{2} \cdot 3^{3} \cdot 41^{2}$ | 6.2.803894544.3 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.108.3t2.b.a | $2$ | $ 2^{2} \cdot 3^{3}$ | 3.1.108.1 | $S_3$ (as 3T2) | $1$ | $0$ |
3.181548.4t5.b.a | $3$ | $ 2^{2} \cdot 3^{3} \cdot 41^{2}$ | 4.2.181548.2 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.13284.6t11.a.a | $3$ | $ 2^{2} \cdot 3^{4} \cdot 41 $ | 6.0.1434672.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
3.544644.6t8.b.a | $3$ | $ 2^{2} \cdot 3^{4} \cdot 41^{2}$ | 4.2.181548.2 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.4428.6t11.a.a | $3$ | $ 2^{2} \cdot 3^{3} \cdot 41 $ | 6.0.1434672.1 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
* | 6.51648192.9t31.a.a | $6$ | $ 2^{6} \cdot 3^{9} \cdot 41 $ | 9.1.5578004736.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |
6.145...112.18t300.a.a | $6$ | $ 2^{6} \cdot 3^{9} \cdot 41^{5}$ | 9.1.5578004736.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.51648192.18t319.a.a | $6$ | $ 2^{6} \cdot 3^{9} \cdot 41 $ | 9.1.5578004736.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.145...112.18t311.a.a | $6$ | $ 2^{6} \cdot 3^{9} \cdot 41^{5}$ | 9.1.5578004736.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.427...984.24t2893.a.a | $8$ | $ 2^{8} \cdot 3^{10} \cdot 41^{4}$ | 9.1.5578004736.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.427...984.12t213.a.a | $8$ | $ 2^{8} \cdot 3^{10} \cdot 41^{4}$ | 9.1.5578004736.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.117...928.36t2217.a.a | $12$ | $ 2^{10} \cdot 3^{15} \cdot 41^{8}$ | 9.1.5578004736.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
12.105...352.36t2214.a.a | $12$ | $ 2^{10} \cdot 3^{17} \cdot 41^{8}$ | 9.1.5578004736.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.753...504.36t2210.a.a | $12$ | $ 2^{12} \cdot 3^{18} \cdot 41^{6}$ | 9.1.5578004736.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.373...032.36t2216.a.a | $12$ | $ 2^{10} \cdot 3^{17} \cdot 41^{4}$ | 9.1.5578004736.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.415...448.18t315.a.a | $12$ | $ 2^{10} \cdot 3^{15} \cdot 41^{4}$ | 9.1.5578004736.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
16.369...184.24t2912.a.a | $16$ | $ 2^{14} \cdot 3^{24} \cdot 41^{8}$ | 9.1.5578004736.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |