Normalized defining polynomial
\( x^{9} - x^{8} + 6x^{7} - 5x^{6} + 7x^{5} + 7x^{4} + 16x^{3} - 3x^{2} - 6x + 2 \)
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: | \(490197028608\) \(\medspace = 2^{8}\cdot 3^{4}\cdot 7^{3}\cdot 41^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(19.90\) | 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: | $2^{3/2}3^{1/2}7^{1/2}41^{1/2}\approx 82.99397568498571$ | ||
Ramified primes: | \(2\), \(3\), \(7\), \(41\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{287}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{38}a^{7}-\frac{7}{38}a^{6}-\frac{15}{38}a^{5}-\frac{3}{19}a^{4}+\frac{5}{38}a^{2}-\frac{7}{19}a-\frac{3}{19}$, $\frac{1}{76}a^{8}-\frac{13}{38}a^{6}-\frac{35}{76}a^{5}-\frac{1}{19}a^{4}-\frac{33}{76}a^{3}-\frac{17}{76}a^{2}-\frac{7}{19}a+\frac{17}{38}$
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: | $\frac{33}{76}a^{8}-\frac{5}{19}a^{7}+\frac{97}{38}a^{6}-\frac{5}{4}a^{5}+\frac{54}{19}a^{4}+\frac{279}{76}a^{3}+\frac{631}{76}a^{2}+\frac{29}{19}a-\frac{63}{38}$, $\frac{7}{76}a^{8}-\frac{3}{19}a^{7}+\frac{27}{38}a^{6}-\frac{65}{76}a^{5}+\frac{30}{19}a^{4}-\frac{3}{76}a^{3}+\frac{125}{76}a^{2}-\frac{45}{19}a+\frac{3}{38}$, $\frac{5}{4}a^{8}-\frac{25}{38}a^{7}+\frac{135}{19}a^{6}-\frac{219}{76}a^{5}+\frac{132}{19}a^{4}+\frac{47}{4}a^{3}+\frac{1935}{76}a^{2}+\frac{118}{19}a-\frac{211}{38}$, $\frac{41}{76}a^{8}-\frac{7}{38}a^{7}+\frac{62}{19}a^{6}+\frac{67}{76}a^{5}+\frac{18}{19}a^{4}+\frac{1079}{76}a^{3}-\frac{159}{76}a^{2}-\frac{86}{19}a+\frac{55}{38}$ | 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: | \( 1001.06570617 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 1001.06570617 \cdot 1}{2\cdot\sqrt{490197028608}}\cr\approx \mathstrut & 2.22841831964 \end{aligned}\]
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
3.1.3444.1 |
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 | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.9.0.1}{9} }$ | R | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
2.4.6.9 | $x^{4} + 2 x^{3} + 6$ | $4$ | $1$ | $6$ | $D_{4}$ | $[2, 2]^{2}$ | |
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.6.3.1 | $x^{6} + 18 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(7\) | 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(41\) | $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
41.2.1.1 | $x^{2} + 41$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
41.2.1.1 | $x^{2} + 41$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
41.2.1.1 | $x^{2} + 41$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.3444.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 7 \cdot 41 $ | \(\Q(\sqrt{-861}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.1148.2t1.a.a | $1$ | $ 2^{2} \cdot 7 \cdot 41 $ | \(\Q(\sqrt{287}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.3444.6t3.l.a | $2$ | $ 2^{2} \cdot 3 \cdot 7 \cdot 41 $ | 6.2.13616584128.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.3444.3t2.a.a | $2$ | $ 2^{2} \cdot 3 \cdot 7 \cdot 41 $ | 3.1.3444.1 | $S_3$ (as 3T2) | $1$ | $0$ |
3.13776.4t5.a.a | $3$ | $ 2^{4} \cdot 3 \cdot 7 \cdot 41 $ | 4.2.13776.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.15814848.6t11.a.a | $3$ | $ 2^{6} \cdot 3 \cdot 7^{2} \cdot 41^{2}$ | 6.0.142333632.5 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
3.47444544.6t8.a.a | $3$ | $ 2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 41^{2}$ | 4.2.13776.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.41328.6t11.a.a | $3$ | $ 2^{4} \cdot 3^{2} \cdot 7 \cdot 41 $ | 6.0.142333632.5 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
* | 6.142333632.9t31.a.a | $6$ | $ 2^{6} \cdot 3^{3} \cdot 7^{2} \cdot 41^{2}$ | 9.1.490197028608.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |
6.187...328.18t300.a.a | $6$ | $ 2^{10} \cdot 3^{3} \cdot 7^{4} \cdot 41^{4}$ | 9.1.490197028608.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.187...328.18t319.a.a | $6$ | $ 2^{10} \cdot 3^{3} \cdot 7^{4} \cdot 41^{4}$ | 9.1.490197028608.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.142333632.18t311.a.a | $6$ | $ 2^{6} \cdot 3^{3} \cdot 7^{2} \cdot 41^{2}$ | 9.1.490197028608.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.296...144.24t2893.a.a | $8$ | $ 2^{16} \cdot 3^{4} \cdot 7^{6} \cdot 41^{6}$ | 9.1.490197028608.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.1708003584.12t213.a.a | $8$ | $ 2^{8} \cdot 3^{4} \cdot 7^{2} \cdot 41^{2}$ | 9.1.490197028608.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.408...744.36t2217.a.a | $12$ | $ 2^{20} \cdot 3^{5} \cdot 7^{7} \cdot 41^{7}$ | 9.1.490197028608.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
12.279...024.36t2214.a.a | $12$ | $ 2^{16} \cdot 3^{7} \cdot 7^{5} \cdot 41^{5}$ | 9.1.490197028608.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.427...736.36t2210.a.a | $12$ | $ 2^{20} \cdot 3^{6} \cdot 7^{6} \cdot 41^{6}$ | 9.1.490197028608.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.367...696.36t2216.a.a | $12$ | $ 2^{20} \cdot 3^{7} \cdot 7^{7} \cdot 41^{7}$ | 9.1.490197028608.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.310...336.18t315.a.a | $12$ | $ 2^{16} \cdot 3^{5} \cdot 7^{5} \cdot 41^{5}$ | 9.1.490197028608.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
16.506...096.24t2912.a.a | $16$ | $ 2^{24} \cdot 3^{8} \cdot 7^{8} \cdot 41^{8}$ | 9.1.490197028608.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |