Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(630745726976\)\(\medspace = 2^{21} \cdot 67^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.660311932928.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 16T1294 |
| Parity: | odd |
| Determinant: | 1.536.2t1.b.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.660311932928.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 12x^{6} - 4x^{5} - 27x^{4} - 24x^{3} + 28x^{2} - 8x + 4 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{3} + 2x + 9 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( a^{2} + 6 a + 7 + \left(10 a^{2} + a + 5\right)\cdot 11 + \left(7 a^{2} + 7 a + 8\right)\cdot 11^{2} + \left(4 a^{2} + 3 a + 2\right)\cdot 11^{3} + \left(6 a^{2} + 2 a + 2\right)\cdot 11^{4} + \left(8 a^{2} + 5 a + 3\right)\cdot 11^{5} + \left(5 a^{2} + 7 a + 7\right)\cdot 11^{6} + \left(8 a^{2} + 7 a + 9\right)\cdot 11^{7} + \left(7 a^{2} + 8 a + 6\right)\cdot 11^{8} + \left(8 a^{2} + 9\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 + 5\cdot 11 + 8\cdot 11^{2} + 3\cdot 11^{3} + 2\cdot 11^{4} + 7\cdot 11^{5} + 7\cdot 11^{6} + 3\cdot 11^{7} + 7\cdot 11^{8} + 8\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a^{2} + 3 a + \left(9 a^{2} + 5 a + 1\right)\cdot 11 + \left(3 a^{2} + 10 a + 3\right)\cdot 11^{2} + \left(a^{2} + 3 a + 9\right)\cdot 11^{3} + \left(5 a^{2} + 10 a + 7\right)\cdot 11^{4} + \left(4 a^{2} + a + 8\right)\cdot 11^{5} + \left(3 a^{2} + 8 a + 7\right)\cdot 11^{6} + \left(6 a^{2} + 10 a + 6\right)\cdot 11^{7} + \left(4 a + 4\right)\cdot 11^{8} + \left(10 a^{2} + 6 a\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a^{2} + 2 a + 10 + \left(2 a^{2} + 4 a + 2\right)\cdot 11 + \left(10 a^{2} + 4 a + 4\right)\cdot 11^{2} + \left(4 a^{2} + 3 a + 10\right)\cdot 11^{3} + \left(10 a^{2} + 9 a + 3\right)\cdot 11^{4} + \left(8 a^{2} + 3 a + 7\right)\cdot 11^{5} + \left(a^{2} + 6 a + 5\right)\cdot 11^{6} + \left(7 a^{2} + 3 a\right)\cdot 11^{7} + \left(2 a^{2} + 8 a\right)\cdot 11^{8} + \left(3 a^{2} + 3 a + 6\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 + 11 + 6\cdot 11^{2} + 10\cdot 11^{3} + 7\cdot 11^{4} + 2\cdot 11^{5} + 11^{6} + 5\cdot 11^{7} + 10\cdot 11^{8} + 5\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a^{2} + 10 a + 4 + \left(3 a^{2} + 4 a + 6\right)\cdot 11 + \left(10 a^{2} + 10 a + 3\right)\cdot 11^{2} + \left(4 a^{2} + 5 a + 5\right)\cdot 11^{3} + \left(a^{2} + 8 a + 8\right)\cdot 11^{4} + \left(5 a^{2} + 3 a\right)\cdot 11^{5} + \left(7 a^{2} + 8 a\right)\cdot 11^{6} + \left(7 a^{2} + 3 a + 9\right)\cdot 11^{7} + \left(6 a + 5\right)\cdot 11^{8} + \left(a^{2} + 3 a + 9\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 5 a^{2} + 3 a + 9 + \left(5 a^{2} + 10 a + 1\right)\cdot 11 + \left(2 a + 5\right)\cdot 11^{2} + \left(2 a^{2} + 5 a + 1\right)\cdot 11^{3} + \left(8 a^{2} + 6 a + 10\right)\cdot 11^{4} + \left(10 a^{2} + 2 a\right)\cdot 11^{5} + \left(4 a^{2} + 4\right)\cdot 11^{6} + \left(6 a^{2} + 5 a + 7\right)\cdot 11^{7} + \left(5 a^{2} + 8 a + 8\right)\cdot 11^{8} + \left(9 a + 8\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 2 a^{2} + 9 a + 5 + \left(2 a^{2} + 6 a + 8\right)\cdot 11 + \left(8 a + 4\right)\cdot 11^{2} + \left(4 a^{2} + 10 a\right)\cdot 11^{3} + \left(a^{2} + 6 a + 1\right)\cdot 11^{4} + \left(6 a^{2} + 4 a + 2\right)\cdot 11^{5} + \left(9 a^{2} + 2 a + 10\right)\cdot 11^{6} + \left(7 a^{2} + 2 a + 1\right)\cdot 11^{7} + \left(4 a^{2} + 7 a\right)\cdot 11^{8} + \left(9 a^{2} + 8 a + 6\right)\cdot 11^{9} +O(11^{10})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(2,7)(6,8)$ | $-3$ |
| $9$ | $2$ | $(1,4)(2,7)(3,5)(6,8)$ | $1$ |
| $12$ | $2$ | $(1,3)$ | $3$ |
| $24$ | $2$ | $(1,2)(3,6)(4,7)(5,8)$ | $3$ |
| $36$ | $2$ | $(1,3)(2,6)$ | $1$ |
| $36$ | $2$ | $(1,3)(2,7)(6,8)$ | $-1$ |
| $16$ | $3$ | $(1,4,5)$ | $0$ |
| $64$ | $3$ | $(1,4,5)(6,7,8)$ | $0$ |
| $12$ | $4$ | $(2,6,7,8)$ | $-3$ |
| $36$ | $4$ | $(1,3,4,5)(2,6,7,8)$ | $1$ |
| $36$ | $4$ | $(1,3,4,5)(2,7)(6,8)$ | $1$ |
| $72$ | $4$ | $(1,2,4,7)(3,6,5,8)$ | $-1$ |
| $72$ | $4$ | $(1,3)(2,6,7,8)$ | $-1$ |
| $144$ | $4$ | $(1,6,3,2)(4,7)(5,8)$ | $1$ |
| $48$ | $6$ | $(1,5,4)(2,7)(6,8)$ | $0$ |
| $96$ | $6$ | $(1,3)(6,8,7)$ | $0$ |
| $192$ | $6$ | $(1,6,4,7,5,8)(2,3)$ | $0$ |
| $144$ | $8$ | $(1,2,3,6,4,7,5,8)$ | $-1$ |
| $96$ | $12$ | $(1,4,5)(2,6,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.