Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(821007451136\)\(\medspace = 2^{10} \cdot 929^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.6.11917436282896.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 16T1294 |
| Parity: | odd |
| Projective image: | $S_4\wr C_2$ |
| Projective field: | Galois closure of 8.6.11917436282896.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$:
\( x^{2} + 127x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 58 a + 75 + \left(112 a + 96\right)\cdot 131 + 35 a\cdot 131^{2} + \left(72 a + 109\right)\cdot 131^{3} + \left(91 a + 43\right)\cdot 131^{4} + \left(109 a + 81\right)\cdot 131^{5} + \left(9 a + 41\right)\cdot 131^{6} + \left(110 a + 17\right)\cdot 131^{7} + \left(16 a + 20\right)\cdot 131^{8} + \left(40 a + 77\right)\cdot 131^{9} +O(131^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 42 + 28\cdot 131 + 56\cdot 131^{2} + 114\cdot 131^{3} + 28\cdot 131^{4} + 54\cdot 131^{5} + 42\cdot 131^{6} + 14\cdot 131^{7} + 32\cdot 131^{8} + 109\cdot 131^{9} +O(131^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 + 29\cdot 131 + 53\cdot 131^{2} + 12\cdot 131^{3} + 95\cdot 131^{4} + 53\cdot 131^{5} + 73\cdot 131^{6} + 111\cdot 131^{7} + 65\cdot 131^{8} + 46\cdot 131^{9} +O(131^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 12 a + 80 + \left(122 a + 60\right)\cdot 131 + \left(28 a + 79\right)\cdot 131^{2} + \left(128 a + 87\right)\cdot 131^{3} + \left(70 a + 56\right)\cdot 131^{4} + \left(73 a + 96\right)\cdot 131^{5} + \left(79 a + 81\right)\cdot 131^{6} + \left(83 a + 71\right)\cdot 131^{7} + \left(34 a + 54\right)\cdot 131^{8} + \left(90 a + 86\right)\cdot 131^{9} +O(131^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 47 a + 108 + \left(20 a + 17\right)\cdot 131 + 59\cdot 131^{2} + \left(31 a + 95\right)\cdot 131^{3} + \left(97 a + 88\right)\cdot 131^{4} + \left(121 a + 8\right)\cdot 131^{5} + \left(115 a + 19\right)\cdot 131^{6} + \left(6 a + 8\right)\cdot 131^{7} + \left(37 a + 127\right)\cdot 131^{8} + \left(45 a + 40\right)\cdot 131^{9} +O(131^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 73 a + 45 + \left(18 a + 95\right)\cdot 131 + \left(95 a + 31\right)\cdot 131^{2} + \left(58 a + 100\right)\cdot 131^{3} + \left(39 a + 75\right)\cdot 131^{4} + \left(21 a + 35\right)\cdot 131^{5} + \left(121 a + 102\right)\cdot 131^{6} + \left(20 a + 54\right)\cdot 131^{7} + \left(114 a + 108\right)\cdot 131^{8} + \left(90 a + 89\right)\cdot 131^{9} +O(131^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 84 a + 34 + \left(110 a + 52\right)\cdot 131 + \left(130 a + 39\right)\cdot 131^{2} + \left(99 a + 88\right)\cdot 131^{3} + \left(33 a + 53\right)\cdot 131^{4} + \left(9 a + 5\right)\cdot 131^{5} + \left(15 a + 99\right)\cdot 131^{6} + \left(124 a + 50\right)\cdot 131^{7} + \left(93 a + 6\right)\cdot 131^{8} + \left(85 a + 54\right)\cdot 131^{9} +O(131^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 119 a + 128 + \left(8 a + 12\right)\cdot 131 + \left(102 a + 73\right)\cdot 131^{2} + \left(2 a + 47\right)\cdot 131^{3} + \left(60 a + 81\right)\cdot 131^{4} + \left(57 a + 57\right)\cdot 131^{5} + \left(51 a + 64\right)\cdot 131^{6} + \left(47 a + 64\right)\cdot 131^{7} + \left(96 a + 109\right)\cdot 131^{8} + \left(40 a + 19\right)\cdot 131^{9} +O(131^{10})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(1,6)(5,7)$ | $-3$ |
| $9$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $1$ |
| $12$ | $2$ | $(2,3)$ | $3$ |
| $24$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $3$ |
| $36$ | $2$ | $(1,5)(2,3)$ | $1$ |
| $36$ | $2$ | $(1,6)(2,3)(5,7)$ | $-1$ |
| $16$ | $3$ | $(2,4,8)$ | $0$ |
| $64$ | $3$ | $(2,4,8)(5,6,7)$ | $0$ |
| $12$ | $4$ | $(1,5,6,7)$ | $-3$ |
| $36$ | $4$ | $(1,5,6,7)(2,3,4,8)$ | $1$ |
| $36$ | $4$ | $(1,6)(2,3,4,8)(5,7)$ | $1$ |
| $72$ | $4$ | $(1,4,6,2)(3,5,8,7)$ | $-1$ |
| $72$ | $4$ | $(1,5,6,7)(2,3)$ | $-1$ |
| $144$ | $4$ | $(1,2,5,3)(4,6)(7,8)$ | $1$ |
| $48$ | $6$ | $(1,6)(2,8,4)(5,7)$ | $0$ |
| $96$ | $6$ | $(2,3)(5,7,6)$ | $0$ |
| $192$ | $6$ | $(1,3)(2,5,4,6,8,7)$ | $0$ |
| $144$ | $8$ | $(1,3,5,4,6,8,7,2)$ | $-1$ |
| $96$ | $12$ | $(1,5,6,7)(2,4,8)$ | $0$ |