Normalized defining polynomial
\( x^{9} - 2x^{8} - 4x^{7} + 11x^{6} + 2x^{5} - 19x^{4} + 7x^{3} + 11x^{2} - 5x - 1 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[5, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(5756350841\) \(\medspace = 17^{4}\cdot 41^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.15\) | 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: | $17^{3/4}41^{1/2}\approx 53.60787834876054$ | ||
Ramified primes: | \(17\), \(41\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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: | $a^{8}-2a^{7}-3a^{6}+8a^{5}+a^{4}-9a^{3}+2a^{2}+3a$, $a^{7}-2a^{6}-2a^{5}+7a^{4}-2a^{3}-6a^{2}+3a+1$, $a^{5}-a^{4}-2a^{3}+3a^{2}-1$, $a^{6}-2a^{5}-2a^{4}+6a^{3}-a^{2}-3a+1$, $a^{5}-a^{4}-3a^{3}+3a^{2}+2a-1$, $a^{2}-1$ | 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: | \( 26.9756599002 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{2}\cdot 26.9756599002 \cdot 1}{2\cdot\sqrt{5756350841}}\cr\approx \mathstrut & 0.224583667190 \end{aligned}\]
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
3.3.697.1 |
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 | ${\href{/padicField/2.9.0.1}{9} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | R | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(17\) | $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
17.4.3.4 | $x^{4} + 102$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(41\) | $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
41.6.3.2 | $x^{6} + 1681 x^{2} - 2412235$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.697.2t1.a.a | $1$ | $ 17 \cdot 41 $ | \(\Q(\sqrt{697}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.41.2t1.a.a | $1$ | $ 41 $ | \(\Q(\sqrt{41}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.697.6t3.b.a | $2$ | $ 17 \cdot 41 $ | 6.6.19918169.1 | $D_{6}$ (as 6T3) | $1$ | $2$ | |
* | 2.697.3t2.a.a | $2$ | $ 17 \cdot 41 $ | 3.3.697.1 | $S_3$ (as 3T2) | $1$ | $2$ |
3.201433.4t5.a.a | $3$ | $ 17^{3} \cdot 41 $ | 4.0.201433.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.11849.6t11.b.a | $3$ | $ 17^{2} \cdot 41 $ | 6.2.8258753.2 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
3.485809.6t8.a.a | $3$ | $ 17^{2} \cdot 41^{2}$ | 4.0.201433.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.8258753.6t11.a.a | $3$ | $ 17^{3} \cdot 41^{2}$ | 6.2.8258753.2 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
* | 6.8258753.9t31.a.a | $6$ | $ 17^{3} \cdot 41^{2}$ | 9.5.5756350841.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ |
6.13882963793.18t300.a.a | $6$ | $ 17^{3} \cdot 41^{4}$ | 9.5.5756350841.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
6.401...177.18t319.a.a | $6$ | $ 17^{5} \cdot 41^{4}$ | 9.5.5756350841.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
6.2386779617.18t311.a.a | $6$ | $ 17^{5} \cdot 41^{2}$ | 9.5.5756350841.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
8.114...129.24t2893.a.a | $8$ | $ 17^{6} \cdot 41^{6}$ | 9.5.5756350841.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.40575253489.12t213.a.a | $8$ | $ 17^{6} \cdot 41^{2}$ | 9.5.5756350841.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.230...857.36t2217.a.a | $12$ | $ 17^{9} \cdot 41^{7}$ | 9.5.5756350841.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.137...897.36t2214.a.a | $12$ | $ 17^{9} \cdot 41^{5}$ | 9.5.5756350841.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.957...209.36t2210.a.a | $12$ | $ 17^{10} \cdot 41^{6}$ | 9.5.5756350841.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-4$ | |
12.230...857.36t2216.a.a | $12$ | $ 17^{9} \cdot 41^{7}$ | 9.5.5756350841.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.137...897.18t315.a.a | $12$ | $ 17^{9} \cdot 41^{5}$ | 9.5.5756350841.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
16.465...081.24t2912.a.a | $16$ | $ 17^{12} \cdot 41^{8}$ | 9.5.5756350841.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |