Normalized defining polynomial
\( x^{9} - x^{8} - 3x^{6} + x^{5} + 3x^{4} + 3x^{3} - 4x^{2} + 1 \)
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: | \(2366667072\) \(\medspace = 2^{6}\cdot 3^{4}\cdot 7^{3}\cdot 11^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.00\) | 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: | $2^{3/2}3^{3/4}7^{1/2}11^{1/2}\approx 56.575877301083764$ | ||
Ramified primes: | \(2\), \(3\), \(7\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{77}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$
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: | $a$, $a^{8}-\frac{2}{3}a^{7}-\frac{10}{3}a^{5}-\frac{1}{3}a^{4}+\frac{7}{3}a^{3}+\frac{14}{3}a^{2}-\frac{5}{3}a-\frac{2}{3}$, $a^{8}-3a^{5}-2a^{4}+a^{3}+4a^{2}$, $\frac{4}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-4a^{5}-\frac{7}{3}a^{4}+\frac{10}{3}a^{3}+\frac{17}{3}a^{2}-\frac{1}{3}$ | 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: | \( 16.5992891922 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 16.5992891922 \cdot 1}{2\cdot\sqrt{2366667072}}\cr\approx \mathstrut & 0.531790262147 \end{aligned}\]
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
3.1.231.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | R | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.9.0.1}{9} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.9.0.1}{9} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
2.6.6.2 | $x^{6} + 6 x^{5} + 14 x^{4} + 24 x^{3} + 44 x^{2} + 8 x + 72$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $[2, 2]^{6}$ | |
\(3\) | 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 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
\(7\) | 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
7.6.3.1 | $x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(11\) | $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
11.6.3.1 | $x^{6} + 242 x^{2} - 11979$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.231.2t1.a.a | $1$ | $ 3 \cdot 7 \cdot 11 $ | \(\Q(\sqrt{-231}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.77.2t1.a.a | $1$ | $ 7 \cdot 11 $ | \(\Q(\sqrt{77}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.231.6t3.f.a | $2$ | $ 3 \cdot 7 \cdot 11 $ | 6.2.4108797.2 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.231.3t2.a.a | $2$ | $ 3 \cdot 7 \cdot 11 $ | 3.1.231.1 | $S_3$ (as 3T2) | $1$ | $0$ |
3.133056.4t5.a.a | $3$ | $ 2^{6} \cdot 3^{3} \cdot 7 \cdot 11 $ | 4.2.133056.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.10245312.6t11.a.a | $3$ | $ 2^{6} \cdot 3^{3} \cdot 7^{2} \cdot 11^{2}$ | 6.0.10245312.4 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
3.3415104.6t8.d.a | $3$ | $ 2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}$ | 4.2.133056.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.44352.6t11.a.a | $3$ | $ 2^{6} \cdot 3^{2} \cdot 7 \cdot 11 $ | 6.0.10245312.4 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
* | 6.10245312.9t31.a.a | $6$ | $ 2^{6} \cdot 3^{3} \cdot 7^{2} \cdot 11^{2}$ | 9.1.2366667072.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |
6.60744454848.18t300.a.a | $6$ | $ 2^{6} \cdot 3^{3} \cdot 7^{4} \cdot 11^{4}$ | 9.1.2366667072.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.546700093632.18t319.a.a | $6$ | $ 2^{6} \cdot 3^{5} \cdot 7^{4} \cdot 11^{4}$ | 9.1.2366667072.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.92207808.18t311.a.a | $6$ | $ 2^{6} \cdot 3^{5} \cdot 7^{2} \cdot 11^{2}$ | 9.1.2366667072.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.622...576.24t2893.b.a | $8$ | $ 2^{12} \cdot 3^{6} \cdot 7^{6} \cdot 11^{6}$ | 9.1.2366667072.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.17703899136.12t213.a.a | $8$ | $ 2^{12} \cdot 3^{6} \cdot 7^{2} \cdot 11^{2}$ | 9.1.2366667072.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.828...256.36t2217.a.a | $12$ | $ 2^{18} \cdot 3^{9} \cdot 7^{7} \cdot 11^{7}$ | 9.1.2366667072.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
12.139...264.36t2214.a.a | $12$ | $ 2^{18} \cdot 3^{9} \cdot 7^{5} \cdot 11^{5}$ | 9.1.2366667072.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.206...976.36t2210.a.a | $12$ | $ 2^{24} \cdot 3^{10} \cdot 7^{6} \cdot 11^{6}$ | 9.1.2366667072.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.828...256.36t2216.a.a | $12$ | $ 2^{18} \cdot 3^{9} \cdot 7^{7} \cdot 11^{7}$ | 9.1.2366667072.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.139...264.18t315.a.a | $12$ | $ 2^{18} \cdot 3^{9} \cdot 7^{5} \cdot 11^{5}$ | 9.1.2366667072.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
16.110...336.24t2912.a.a | $16$ | $ 2^{24} \cdot 3^{12} \cdot 7^{8} \cdot 11^{8}$ | 9.1.2366667072.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |