Normalized defining polynomial
\( x^{9} - 4x^{8} + 4x^{7} + 3x^{6} - 2x^{5} - 7x^{4} + 11x^{2} + 4x + 1 \)
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: | \(77145562593\) \(\medspace = 3^{6}\cdot 11^{3}\cdot 43^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.21\) | 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: | $3^{43/36}11^{1/2}43^{1/2}\approx 80.78400912272974$ | ||
Ramified primes: | \(3\), \(11\), \(43\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{473}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{43}a^{8}+\frac{7}{43}a^{7}-\frac{5}{43}a^{6}-\frac{9}{43}a^{5}-\frac{15}{43}a^{4}+\frac{11}{43}a-\frac{4}{43}$
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: | $a^{8}-4a^{7}+4a^{6}+3a^{5}-2a^{4}-7a^{3}+11a+4$, $\frac{15}{43}a^{8}-\frac{67}{43}a^{7}+\frac{97}{43}a^{6}-\frac{6}{43}a^{5}-\frac{53}{43}a^{4}-a^{3}+a^{2}+\frac{122}{43}a-\frac{17}{43}$, $\frac{7}{43}a^{8}-\frac{37}{43}a^{7}+\frac{51}{43}a^{6}+\frac{23}{43}a^{5}-\frac{62}{43}a^{4}-a^{3}+a^{2}+\frac{120}{43}a-\frac{28}{43}$, $\frac{90}{43}a^{8}-\frac{359}{43}a^{7}+\frac{367}{43}a^{6}+\frac{265}{43}a^{5}-\frac{232}{43}a^{4}-14a^{3}+2a^{2}+\frac{1033}{43}a+\frac{285}{43}$ | 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: | \( 105.580383931 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 105.580383931 \cdot 1}{2\cdot\sqrt{77145562593}}\cr\approx \mathstrut & 0.592443903959 \end{aligned}\]
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
3.1.1419.1 |
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 | ${\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.9.0.1}{9} }$ | R | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | R | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
3.6.6.7 | $x^{6} + 3 x + 6$ | $6$ | $1$ | $6$ | $C_3^2:D_4$ | $[5/4, 5/4]_{4}^{2}$ | |
\(11\) | $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(43\) | 43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
43.3.0.1 | $x^{3} + x + 40$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.1419.2t1.a.a | $1$ | $ 3 \cdot 11 \cdot 43 $ | \(\Q(\sqrt{-1419}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.473.2t1.a.a | $1$ | $ 11 \cdot 43 $ | \(\Q(\sqrt{473}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.1419.6t3.c.a | $2$ | $ 3 \cdot 11 \cdot 43 $ | 6.2.952414353.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.1419.3t2.a.a | $2$ | $ 3 \cdot 11 \cdot 43 $ | 3.1.1419.1 | $S_3$ (as 3T2) | $1$ | $0$ |
3.12771.4t5.a.a | $3$ | $ 3^{3} \cdot 11 \cdot 43 $ | 4.2.12771.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.4257.6t11.a.a | $3$ | $ 3^{2} \cdot 11 \cdot 43 $ | 6.0.6040683.2 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
3.2013561.6t8.c.a | $3$ | $ 3^{2} \cdot 11^{2} \cdot 43^{2}$ | 4.2.12771.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.6040683.6t11.a.a | $3$ | $ 3^{3} \cdot 11^{2} \cdot 43^{2}$ | 6.0.6040683.2 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
* | 6.54366147.9t31.a.a | $6$ | $ 3^{5} \cdot 11^{2} \cdot 43^{2}$ | 9.1.77145562593.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |
6.121...163.18t300.a.a | $6$ | $ 3^{5} \cdot 11^{4} \cdot 43^{4}$ | 9.1.77145562593.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.109...467.18t319.a.a | $6$ | $ 3^{7} \cdot 11^{4} \cdot 43^{4}$ | 9.1.77145562593.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.489295323.18t311.a.a | $6$ | $ 3^{7} \cdot 11^{2} \cdot 43^{2}$ | 9.1.77145562593.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.661...961.24t2893.a.a | $8$ | $ 3^{10} \cdot 11^{6} \cdot 43^{6}$ | 9.1.77145562593.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.13210973721.12t213.a.a | $8$ | $ 3^{10} \cdot 11^{2} \cdot 43^{2}$ | 9.1.77145562593.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.760...379.36t2217.a.a | $12$ | $ 3^{15} \cdot 11^{7} \cdot 43^{7}$ | 9.1.77145562593.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
12.339...851.36t2214.a.a | $12$ | $ 3^{15} \cdot 11^{5} \cdot 43^{5}$ | 9.1.77145562593.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.535...841.36t2210.a.a | $12$ | $ 3^{14} \cdot 11^{6} \cdot 43^{6}$ | 9.1.77145562593.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.760...379.36t2216.a.a | $12$ | $ 3^{15} \cdot 11^{7} \cdot 43^{7}$ | 9.1.77145562593.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.339...851.18t315.a.a | $12$ | $ 3^{15} \cdot 11^{5} \cdot 43^{5}$ | 9.1.77145562593.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
16.873...881.24t2912.a.a | $16$ | $ 3^{20} \cdot 11^{8} \cdot 43^{8}$ | 9.1.77145562593.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |