Normalized defining polynomial
\( x^{9} - 3x^{8} + 4x^{7} + 2x^{5} - 2x^{4} - 2x^{3} + 4x^{2} + x + 1 \)
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: | \(86537154816\) \(\medspace = 2^{8}\cdot 3^{4}\cdot 7^{3}\cdot 23^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.42\) | 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: | $2^{8/9}3^{3/4}7^{1/2}23^{1/2}\approx 53.5594486205667$ | ||
Ramified primes: | \(2\), \(3\), \(7\), \(23\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{161}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5}a^{7}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}$, $\frac{1}{5}a^{8}-\frac{2}{5}a^{6}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}a+\frac{2}{5}$
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: | $a^{8}-3a^{7}+4a^{6}+2a^{4}-2a^{3}-2a^{2}+4a+1$, $\frac{3}{5}a^{8}-2a^{7}+\frac{14}{5}a^{6}-\frac{1}{5}a^{4}-\frac{2}{5}a^{3}-\frac{8}{5}a^{2}+\frac{21}{5}a-\frac{4}{5}$, $\frac{2}{5}a^{8}-\frac{9}{5}a^{7}+3a^{6}-\frac{6}{5}a^{5}-\frac{3}{5}a^{4}-\frac{11}{5}a^{3}-\frac{3}{5}a^{2}+\frac{19}{5}a+\frac{1}{5}$, $a^{8}-\frac{11}{5}a^{7}+\frac{6}{5}a^{6}+\frac{21}{5}a^{5}+\frac{4}{5}a^{4}-\frac{12}{5}a^{3}-\frac{14}{5}a^{2}+4a+\frac{18}{5}$ | 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: | \( 205.305820981 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 205.305820981 \cdot 1}{2\cdot\sqrt{86537154816}}\cr\approx \mathstrut & 1.08772580123 \end{aligned}\]
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
3.1.1932.1 |
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 | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.9.0.1}{9} }$ | R | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.9.8.1 | $x^{9} + 2$ | $9$ | $1$ | $8$ | $(C_9:C_3):C_2$ | $[\ ]_{9}^{6}$ |
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
3.4.3.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
\(7\) | 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(23\) | $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.6.3.1 | $x^{6} + 1058 x^{2} - 219006$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.483.2t1.a.a | $1$ | $ 3 \cdot 7 \cdot 23 $ | \(\Q(\sqrt{-483}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.161.2t1.a.a | $1$ | $ 7 \cdot 23 $ | \(\Q(\sqrt{161}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.1932.6t3.f.a | $2$ | $ 2^{2} \cdot 3 \cdot 7 \cdot 23 $ | 6.2.600952464.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.1932.3t2.a.a | $2$ | $ 2^{2} \cdot 3 \cdot 7 \cdot 23 $ | 3.1.1932.1 | $S_3$ (as 3T2) | $1$ | $0$ |
3.17388.4t5.a.a | $3$ | $ 2^{2} \cdot 3^{3} \cdot 7 \cdot 23 $ | 4.2.17388.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.5796.6t11.a.a | $3$ | $ 2^{2} \cdot 3^{2} \cdot 7 \cdot 23 $ | 6.0.11197872.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
3.933156.6t8.a.a | $3$ | $ 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 23^{2}$ | 4.2.17388.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.2799468.6t11.a.a | $3$ | $ 2^{2} \cdot 3^{3} \cdot 7^{2} \cdot 23^{2}$ | 6.0.11197872.1 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
* | 6.44791488.9t31.b.a | $6$ | $ 2^{6} \cdot 3^{3} \cdot 7^{2} \cdot 23^{2}$ | 9.1.86537154816.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |
6.116...448.18t300.a.a | $6$ | $ 2^{6} \cdot 3^{3} \cdot 7^{4} \cdot 23^{4}$ | 9.1.86537154816.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.104...032.18t319.a.a | $6$ | $ 2^{6} \cdot 3^{5} \cdot 7^{4} \cdot 23^{4}$ | 9.1.86537154816.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.403123392.18t311.a.a | $6$ | $ 2^{6} \cdot 3^{5} \cdot 7^{2} \cdot 23^{2}$ | 9.1.86537154816.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.325...664.24t2893.a.a | $8$ | $ 2^{8} \cdot 3^{6} \cdot 7^{6} \cdot 23^{6}$ | 9.1.86537154816.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.4837480704.12t213.a.a | $8$ | $ 2^{8} \cdot 3^{6} \cdot 7^{2} \cdot 23^{2}$ | 9.1.86537154816.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.565...632.36t2217.a.a | $12$ | $ 2^{10} \cdot 3^{9} \cdot 7^{7} \cdot 23^{7}$ | 9.1.86537154816.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
12.218...992.36t2214.a.a | $12$ | $ 2^{10} \cdot 3^{9} \cdot 7^{5} \cdot 23^{5}$ | 9.1.86537154816.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.421...544.36t2210.a.a | $12$ | $ 2^{12} \cdot 3^{10} \cdot 7^{6} \cdot 23^{6}$ | 9.1.86537154816.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.565...632.36t2216.a.a | $12$ | $ 2^{10} \cdot 3^{9} \cdot 7^{7} \cdot 23^{7}$ | 9.1.86537154816.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.218...992.18t315.a.a | $12$ | $ 2^{10} \cdot 3^{9} \cdot 7^{5} \cdot 23^{5}$ | 9.1.86537154816.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
16.393...864.24t2912.a.a | $16$ | $ 2^{14} \cdot 3^{12} \cdot 7^{8} \cdot 23^{8}$ | 9.1.86537154816.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |