Normalized defining polynomial
\( x^{9} - 2x^{8} - 5x^{7} + 10x^{6} + 9x^{5} - 19x^{4} - 6x^{3} + 17x^{2} + 2x - 4 \)
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: | \(-17545148927\) \(\medspace = -\,47\cdot 139^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.75\) | 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: | $47^{1/2}139^{1/2}\approx 80.82697569499925$ | ||
Ramified primes: | \(47\), \(139\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-47}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$
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: | $\frac{1}{2}a^{8}-a^{7}-\frac{5}{2}a^{6}+5a^{5}+\frac{7}{2}a^{4}-\frac{17}{2}a^{3}+\frac{13}{2}a-1$, $a^{3}-a^{2}-2a+1$, $\frac{1}{2}a^{8}-a^{7}-\frac{5}{2}a^{6}+5a^{5}+\frac{9}{2}a^{4}-\frac{19}{2}a^{3}-3a^{2}+\frac{15}{2}a+2$, $\frac{1}{2}a^{8}-a^{7}-\frac{3}{2}a^{6}+3a^{5}+\frac{3}{2}a^{4}-\frac{7}{2}a^{3}+\frac{5}{2}a+1$, $a+1$ | 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: | \( 60.635100851 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 60.635100851 \cdot 1}{2\cdot\sqrt{17545148927}}\cr\approx \mathstrut & 0.45419795152 \end{aligned}\]
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
3.1.139.1 |
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 | ${\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.3.0.1}{3} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | R | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(47\) | $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.3.0.1 | $x^{3} + 3 x + 42$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(139\) | $\Q_{139}$ | $x + 137$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
139.2.1.1 | $x^{2} + 278$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
139.6.3.2 | $x^{6} + 429 x^{4} + 274 x^{3} + 57999 x^{2} - 112614 x + 2477540$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.139.2t1.a.a | $1$ | $ 139 $ | \(\Q(\sqrt{-139}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.6533.2t1.a.a | $1$ | $ 47 \cdot 139 $ | \(\Q(\sqrt{6533}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.47.2t1.a.a | $1$ | $ 47 $ | \(\Q(\sqrt{-47}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.307051.6t3.a.a | $2$ | $ 47^{2} \cdot 139 $ | 6.0.2005964183.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.139.3t2.a.a | $2$ | $ 139 $ | 3.1.139.1 | $S_3$ (as 3T2) | $1$ | $0$ |
3.307051.4t5.a.a | $3$ | $ 47^{2} \cdot 139 $ | 4.2.307051.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.908087.6t11.a.a | $3$ | $ 47 \cdot 139^{2}$ | 6.2.126224093.1 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
3.42680089.6t8.a.a | $3$ | $ 47^{2} \cdot 139^{2}$ | 4.2.307051.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.6533.6t11.a.a | $3$ | $ 47 \cdot 139 $ | 6.2.126224093.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
* | 6.126224093.9t31.a.a | $6$ | $ 47 \cdot 139^{3}$ | 9.3.17545148927.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ |
6.615...333.18t300.a.a | $6$ | $ 47^{5} \cdot 139^{3}$ | 9.3.17545148927.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
6.126224093.18t319.a.a | $6$ | $ 47 \cdot 139^{3}$ | 9.3.17545148927.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
6.615...333.18t311.a.a | $6$ | $ 47^{5} \cdot 139^{3}$ | 9.3.17545148927.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
8.182...921.24t2893.a.a | $8$ | $ 47^{4} \cdot 139^{4}$ | 9.3.17545148927.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-4$ | |
8.182...921.12t213.a.a | $8$ | $ 47^{4} \cdot 139^{4}$ | 9.3.17545148927.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $4$ | |
12.123...939.36t2217.a.a | $12$ | $ 47^{8} \cdot 139^{5}$ | 9.3.17545148927.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.238...419.36t2214.a.a | $12$ | $ 47^{8} \cdot 139^{7}$ | 9.3.17545148927.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
12.777...969.36t2210.a.a | $12$ | $ 47^{6} \cdot 139^{6}$ | 9.3.17545148927.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.489...099.36t2216.a.a | $12$ | $ 47^{4} \cdot 139^{7}$ | 9.3.17545148927.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.253...019.18t315.a.a | $12$ | $ 47^{4} \cdot 139^{5}$ | 9.3.17545148927.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
16.331...241.24t2912.a.a | $16$ | $ 47^{8} \cdot 139^{8}$ | 9.3.17545148927.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |