Basic invariants
| Dimension: | $6$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(106168320000\)\(\medspace = 2^{21} \cdot 3^{4} \cdot 5^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.2.5898240000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T200 |
| Parity: | even |
| Projective image: | $S_4\wr C_2$ |
| Projective field: | Galois closure of 8.2.5898240000.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 167 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 167 }$:
\( x^{2} + 166x + 5 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 124 a + 90 + \left(160 a + 147\right)\cdot 167 + \left(116 a + 88\right)\cdot 167^{2} + \left(113 a + 27\right)\cdot 167^{3} + \left(163 a + 102\right)\cdot 167^{4} + \left(120 a + 27\right)\cdot 167^{5} + \left(162 a + 10\right)\cdot 167^{6} + \left(164 a + 28\right)\cdot 167^{7} + \left(2 a + 15\right)\cdot 167^{8} + \left(51 a + 24\right)\cdot 167^{9} +O(167^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 32 + 63\cdot 167 + 161\cdot 167^{2} + 19\cdot 167^{3} + 127\cdot 167^{4} + 106\cdot 167^{5} + 141\cdot 167^{6} + 120\cdot 167^{7} + 41\cdot 167^{8} + 47\cdot 167^{9} +O(167^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 43 a + 68 + \left(38 a + 117\right)\cdot 167 + \left(5 a + 142\right)\cdot 167^{2} + \left(96 a + 140\right)\cdot 167^{3} + \left(114 a + 160\right)\cdot 167^{4} + \left(89 a + 130\right)\cdot 167^{5} + \left(48 a + 106\right)\cdot 167^{6} + \left(85 a + 95\right)\cdot 167^{7} + \left(133 a + 63\right)\cdot 167^{8} + \left(37 a + 155\right)\cdot 167^{9} +O(167^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 43 a + 47 + \left(6 a + 17\right)\cdot 167 + \left(50 a + 45\right)\cdot 167^{2} + \left(53 a + 24\right)\cdot 167^{3} + \left(3 a + 152\right)\cdot 167^{4} + \left(46 a + 151\right)\cdot 167^{5} + \left(4 a + 51\right)\cdot 167^{6} + \left(2 a + 30\right)\cdot 167^{7} + \left(164 a + 20\right)\cdot 167^{8} + \left(115 a + 72\right)\cdot 167^{9} +O(167^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 124 a + 111 + \left(128 a + 112\right)\cdot 167 + \left(161 a + 109\right)\cdot 167^{2} + \left(70 a + 64\right)\cdot 167^{3} + \left(52 a + 12\right)\cdot 167^{4} + \left(77 a + 106\right)\cdot 167^{5} + \left(118 a + 65\right)\cdot 167^{6} + \left(81 a + 132\right)\cdot 167^{7} + \left(33 a + 111\right)\cdot 167^{8} + \left(129 a + 59\right)\cdot 167^{9} +O(167^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 165 + 105\cdot 167 + 38\cdot 167^{2} + 95\cdot 167^{3} + 119\cdot 167^{4} + 47\cdot 167^{5} + 130\cdot 167^{6} + 154\cdot 167^{7} + 89\cdot 167^{8} + 23\cdot 167^{9} +O(167^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 57 a + 49 + \left(110 a + 25\right)\cdot 167 + \left(61 a + 65\right)\cdot 167^{2} + \left(39 a + 75\right)\cdot 167^{3} + \left(4 a + 14\right)\cdot 167^{4} + \left(118 a + 75\right)\cdot 167^{5} + \left(99 a + 6\right)\cdot 167^{6} + \left(21 a + 92\right)\cdot 167^{7} + \left(11 a + 84\right)\cdot 167^{8} + \left(162 a + 67\right)\cdot 167^{9} +O(167^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 110 a + 106 + \left(56 a + 78\right)\cdot 167 + \left(105 a + 16\right)\cdot 167^{2} + \left(127 a + 53\right)\cdot 167^{3} + \left(162 a + 146\right)\cdot 167^{4} + \left(48 a + 21\right)\cdot 167^{5} + \left(67 a + 155\right)\cdot 167^{6} + \left(145 a + 13\right)\cdot 167^{7} + \left(155 a + 74\right)\cdot 167^{8} + \left(4 a + 51\right)\cdot 167^{9} +O(167^{10})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $6$ |
| $6$ | $2$ | $(1,4)(2,6)$ | $2$ |
| $9$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $-2$ |
| $12$ | $2$ | $(3,5)$ | $2$ |
| $24$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $36$ | $2$ | $(1,2)(3,5)$ | $-2$ |
| $36$ | $2$ | $(1,4)(2,6)(3,5)$ | $-2$ |
| $16$ | $3$ | $(3,7,8)$ | $3$ |
| $64$ | $3$ | $(2,4,6)(3,7,8)$ | $0$ |
| $12$ | $4$ | $(1,2,4,6)$ | $4$ |
| $36$ | $4$ | $(1,2,4,6)(3,5,7,8)$ | $2$ |
| $36$ | $4$ | $(1,4)(2,6)(3,5,7,8)$ | $0$ |
| $72$ | $4$ | $(1,7,4,3)(2,8,6,5)$ | $0$ |
| $72$ | $4$ | $(1,2,4,6)(3,5)$ | $0$ |
| $144$ | $4$ | $(1,3,2,5)(4,7)(6,8)$ | $0$ |
| $48$ | $6$ | $(1,4)(2,6)(3,8,7)$ | $-1$ |
| $96$ | $6$ | $(2,6,4)(3,5)$ | $-1$ |
| $192$ | $6$ | $(1,5)(2,7,4,8,6,3)$ | $0$ |
| $144$ | $8$ | $(1,5,2,7,4,8,6,3)$ | $0$ |
| $96$ | $12$ | $(1,2,4,6)(3,7,8)$ | $1$ |