Normalized defining polynomial
\( x^{9} + 4x^{7} - 6x^{6} + 13x^{5} - 16x^{4} + 18x^{3} - 26x^{2} + 16x - 8 \)
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: | \(26357170176\) \(\medspace = 2^{15}\cdot 3^{3}\cdot 31^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(14.38\) | 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: | $2^{17/6}3^{1/2}31^{1/2}\approx 68.73212880929385$ | ||
Ramified primes: | \(2\), \(3\), \(31\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{186}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{12}a^{7}+\frac{1}{12}a^{6}+\frac{1}{12}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{3}a^{2}+\frac{1}{6}a+\frac{1}{3}$, $\frac{1}{72}a^{8}+\frac{1}{36}a^{7}+\frac{1}{9}a^{6}+\frac{5}{36}a^{5}+\frac{11}{24}a^{4}-\frac{11}{36}a^{3}-\frac{13}{36}a^{2}-\frac{1}{12}a+\frac{1}{18}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | $\frac{1}{24}a^{8}+\frac{1}{4}a^{6}-\frac{1}{6}a^{5}+\frac{5}{8}a^{4}-\frac{11}{12}a^{3}+\frac{1}{4}a^{2}-\frac{5}{12}a-\frac{1}{6}$, $\frac{7}{36}a^{8}-\frac{1}{9}a^{7}+\frac{5}{9}a^{6}-\frac{14}{9}a^{5}+\frac{29}{12}a^{4}-\frac{59}{18}a^{3}+\frac{53}{18}a^{2}-\frac{25}{6}a+\frac{34}{9}$, $\frac{11}{36}a^{8}+\frac{4}{9}a^{7}+\frac{23}{18}a^{6}-\frac{1}{9}a^{5}+\frac{19}{12}a^{4}+\frac{5}{18}a^{3}-\frac{5}{18}a^{2}-\frac{13}{6}a-\frac{40}{9}$, $\frac{7}{24}a^{8}+\frac{1}{4}a^{7}+a^{6}-\frac{11}{12}a^{5}+\frac{13}{8}a^{4}-\frac{23}{12}a^{3}+\frac{7}{4}a^{2}-\frac{41}{12}a-\frac{1}{6}$ | 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: | \( 225.473878388 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 225.473878388 \cdot 1}{2\cdot\sqrt{26357170176}}\cr\approx \mathstrut & 2.16454350127 \end{aligned}\]
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
3.1.744.1 |
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 | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.9.0.1}{9} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | R | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.4.10.5 | $x^{4} + 4 x^{3} + 4 x^{2} + 10$ | $4$ | $1$ | $10$ | $D_{4}$ | $[2, 3, 7/2]$ | |
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
3.6.3.1 | $x^{6} + 18 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(31\) | $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
31.2.1.1 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.744.2t1.b.a | $1$ | $ 2^{3} \cdot 3 \cdot 31 $ | \(\Q(\sqrt{-186}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.744.2t1.a.a | $1$ | $ 2^{3} \cdot 3 \cdot 31 $ | \(\Q(\sqrt{186}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.2976.6t3.c.a | $2$ | $ 2^{5} \cdot 3 \cdot 31 $ | 6.2.1647323136.2 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.744.3t2.a.a | $2$ | $ 2^{3} \cdot 3 \cdot 31 $ | 3.1.744.1 | $S_3$ (as 3T2) | $1$ | $0$ |
3.47616.4t5.a.a | $3$ | $ 2^{9} \cdot 3 \cdot 31 $ | 4.2.47616.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.8856576.6t11.b.a | $3$ | $ 2^{10} \cdot 3^{2} \cdot 31^{2}$ | 6.0.8856576.2 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
3.8856576.6t8.a.a | $3$ | $ 2^{10} \cdot 3^{2} \cdot 31^{2}$ | 4.2.47616.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.11904.6t11.b.a | $3$ | $ 2^{7} \cdot 3 \cdot 31 $ | 6.0.8856576.2 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
* | 6.35426304.9t31.a.a | $6$ | $ 2^{12} \cdot 3^{2} \cdot 31^{2}$ | 9.1.26357170176.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |
6.196...944.18t300.a.a | $6$ | $ 2^{18} \cdot 3^{4} \cdot 31^{4}$ | 9.1.26357170176.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.490...736.18t319.a.a | $6$ | $ 2^{16} \cdot 3^{4} \cdot 31^{4}$ | 9.1.26357170176.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.566820864.18t311.a.a | $6$ | $ 2^{16} \cdot 3^{2} \cdot 31^{2}$ | 9.1.26357170176.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.434...936.24t2893.a.a | $8$ | $ 2^{26} \cdot 3^{6} \cdot 31^{6}$ | 9.1.26357170176.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.9069133824.12t213.a.a | $8$ | $ 2^{20} \cdot 3^{2} \cdot 31^{2}$ | 9.1.26357170176.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.826...304.36t2217.a.a | $12$ | $ 2^{37} \cdot 3^{7} \cdot 31^{7}$ | 9.1.26357170176.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
12.597...256.36t2214.a.a | $12$ | $ 2^{33} \cdot 3^{5} \cdot 31^{5}$ | 9.1.26357170176.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.444...464.36t2210.a.a | $12$ | $ 2^{36} \cdot 3^{6} \cdot 31^{6}$ | 9.1.26357170176.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.206...576.36t2216.a.a | $12$ | $ 2^{35} \cdot 3^{7} \cdot 31^{7}$ | 9.1.26357170176.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.149...064.18t315.a.a | $12$ | $ 2^{31} \cdot 3^{5} \cdot 31^{5}$ | 9.1.26357170176.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
16.393...264.24t2912.a.a | $16$ | $ 2^{46} \cdot 3^{8} \cdot 31^{8}$ | 9.1.26357170176.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |