Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(729212883968\)\(\medspace = 2^{10} \cdot 19^{3} \cdot 47^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.94939073536.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 16T1294 |
| Parity: | odd |
| Determinant: | 1.3572.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.94939073536.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 2x^{6} - 2x^{5} + 37x^{4} - 42x^{3} - 62x^{2} + 116x - 44 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 22 a + 18 + \left(a + 34\right)\cdot 37 + \left(24 a + 21\right)\cdot 37^{2} + 28\cdot 37^{3} + \left(17 a + 3\right)\cdot 37^{4} + \left(5 a + 27\right)\cdot 37^{5} + \left(23 a + 18\right)\cdot 37^{6} + \left(14 a + 5\right)\cdot 37^{7} + 34\cdot 37^{8} + \left(36 a + 5\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 a + 34 + \left(17 a + 16\right)\cdot 37 + \left(29 a + 28\right)\cdot 37^{2} + \left(32 a + 23\right)\cdot 37^{3} + \left(10 a + 12\right)\cdot 37^{4} + 6 a\cdot 37^{5} + \left(30 a + 10\right)\cdot 37^{6} + \left(19 a + 26\right)\cdot 37^{7} + \left(2 a + 32\right)\cdot 37^{8} + \left(6 a + 3\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 a + 32 + \left(35 a + 18\right)\cdot 37 + \left(12 a + 5\right)\cdot 37^{2} + \left(36 a + 7\right)\cdot 37^{3} + \left(19 a + 34\right)\cdot 37^{4} + \left(31 a + 31\right)\cdot 37^{5} + \left(13 a + 31\right)\cdot 37^{6} + \left(22 a + 3\right)\cdot 37^{7} + \left(36 a + 21\right)\cdot 37^{8} + 37^{9} +O(37^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( a + 14 + \left(23 a + 19\right)\cdot 37 + \left(17 a + 31\right)\cdot 37^{2} + \left(a + 6\right)\cdot 37^{3} + \left(34 a + 12\right)\cdot 37^{4} + \left(34 a + 29\right)\cdot 37^{5} + \left(3 a + 24\right)\cdot 37^{6} + \left(24 a + 26\right)\cdot 37^{7} + \left(7 a + 22\right)\cdot 37^{8} + \left(33 a + 7\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 23 + 11\cdot 37 + 25\cdot 37^{2} + 4\cdot 37^{3} + 9\cdot 37^{4} + 34\cdot 37^{6} + 22\cdot 37^{7} + 23\cdot 37^{8} + 29\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 21 + 6\cdot 37 + 12\cdot 37^{2} + 30\cdot 37^{3} + 16\cdot 37^{4} + 20\cdot 37^{5} + 9\cdot 37^{6} + 16\cdot 37^{7} + 35\cdot 37^{8} + 14\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 36 a + 18 + \left(13 a + 36\right)\cdot 37 + \left(19 a + 4\right)\cdot 37^{2} + \left(35 a + 32\right)\cdot 37^{3} + \left(2 a + 35\right)\cdot 37^{4} + \left(2 a + 23\right)\cdot 37^{5} + \left(33 a + 5\right)\cdot 37^{6} + \left(12 a + 8\right)\cdot 37^{7} + \left(29 a + 29\right)\cdot 37^{8} + \left(3 a + 21\right)\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 29 a + 29 + \left(19 a + 3\right)\cdot 37 + \left(7 a + 18\right)\cdot 37^{2} + \left(4 a + 14\right)\cdot 37^{3} + \left(26 a + 23\right)\cdot 37^{4} + \left(30 a + 14\right)\cdot 37^{5} + \left(6 a + 13\right)\cdot 37^{6} + \left(17 a + 1\right)\cdot 37^{7} + \left(34 a + 23\right)\cdot 37^{8} + \left(30 a + 25\right)\cdot 37^{9} +O(37^{10})\)
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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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(1,3)(2,8)$ | $-3$ |
| $9$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | $1$ |
| $12$ | $2$ | $(1,2)$ | $3$ |
| $24$ | $2$ | $(1,4)(2,5)(3,6)(7,8)$ | $3$ |
| $36$ | $2$ | $(1,2)(4,5)$ | $1$ |
| $36$ | $2$ | $(1,2)(4,6)(5,7)$ | $-1$ |
| $16$ | $3$ | $(1,3,8)$ | $0$ |
| $64$ | $3$ | $(1,3,8)(4,6,7)$ | $0$ |
| $12$ | $4$ | $(1,2,3,8)$ | $-3$ |
| $36$ | $4$ | $(1,2,3,8)(4,5,6,7)$ | $1$ |
| $36$ | $4$ | $(1,3)(2,8)(4,5,6,7)$ | $1$ |
| $72$ | $4$ | $(1,6,3,4)(2,7,8,5)$ | $-1$ |
| $72$ | $4$ | $(1,2)(4,5,6,7)$ | $-1$ |
| $144$ | $4$ | $(1,5,2,4)(3,6)(7,8)$ | $1$ |
| $48$ | $6$ | $(1,8,3)(4,6)(5,7)$ | $0$ |
| $96$ | $6$ | $(1,2)(5,7,6)$ | $0$ |
| $192$ | $6$ | $(1,6,3,7,8,4)(2,5)$ | $0$ |
| $144$ | $8$ | $(1,5,2,6,3,7,8,4)$ | $-1$ |
| $96$ | $12$ | $(1,3,8)(4,5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.