Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(341\!\cdots\!000\)\(\medspace = 2^{36} \cdot 3^{26} \cdot 5^{9} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.116640000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | odd |
| Determinant: | 1.20.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.116640000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 10x^{6} - 14x^{5} + 14x^{4} - 8x^{3} - x^{2} + 4x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 149 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 149 }$:
\( x^{3} + 3x + 147 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 92 + 143\cdot 149 + 95\cdot 149^{2} + 8\cdot 149^{3} + 140\cdot 149^{4} + 116\cdot 149^{5} + 46\cdot 149^{6} + 34\cdot 149^{7} + 113\cdot 149^{8} + 17\cdot 149^{9} +O(149^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 111 + 104\cdot 149 + 61\cdot 149^{2} + 52\cdot 149^{3} + 138\cdot 149^{4} + 20\cdot 149^{5} + 100\cdot 149^{6} + 65\cdot 149^{7} + 38\cdot 149^{8} + 61\cdot 149^{9} +O(149^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 42 a^{2} + 37 a + 54 + \left(121 a^{2} + 112 a + 95\right)\cdot 149 + \left(62 a^{2} + 12 a + 93\right)\cdot 149^{2} + \left(62 a^{2} + 79 a + 22\right)\cdot 149^{3} + \left(117 a^{2} + 2 a + 39\right)\cdot 149^{4} + \left(133 a^{2} + 79 a + 129\right)\cdot 149^{5} + \left(13 a^{2} + 38 a + 61\right)\cdot 149^{6} + \left(34 a^{2} + 129 a + 106\right)\cdot 149^{7} + \left(107 a^{2} + 130 a + 27\right)\cdot 149^{8} + \left(39 a^{2} + 127 a + 123\right)\cdot 149^{9} +O(149^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 44 a^{2} + 25 a + 2 + \left(27 a^{2} + 96 a + 119\right)\cdot 149 + \left(112 a^{2} + 112 a + 54\right)\cdot 149^{2} + \left(143 a^{2} + 45 a + 71\right)\cdot 149^{3} + \left(130 a^{2} + 94 a + 116\right)\cdot 149^{4} + \left(109 a^{2} + 113 a + 63\right)\cdot 149^{5} + \left(146 a^{2} + 21 a + 61\right)\cdot 149^{6} + \left(141 a^{2} + 33 a + 63\right)\cdot 149^{7} + \left(130 a^{2} + 133 a + 50\right)\cdot 149^{8} + \left(24 a^{2} + 128 a + 29\right)\cdot 149^{9} +O(149^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 115 a^{2} + 35 a + 144 + \left(24 a^{2} + 82 a + 113\right)\cdot 149 + \left(20 a^{2} + 4 a + 19\right)\cdot 149^{2} + \left(58 a^{2} + 115 a + 49\right)\cdot 149^{3} + \left(83 a^{2} + 137 a + 21\right)\cdot 149^{4} + \left(89 a^{2} + 9 a + 23\right)\cdot 149^{5} + \left(91 a^{2} + 25 a + 100\right)\cdot 149^{6} + \left(50 a^{2} + 12 a + 29\right)\cdot 149^{7} + \left(59 a^{2} + 123 a + 56\right)\cdot 149^{8} + \left(123 a^{2} + 140 a + 77\right)\cdot 149^{9} +O(149^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 117 a^{2} + 51 a + 55 + \left(131 a^{2} + 60 a + 116\right)\cdot 149 + \left(37 a^{2} + 108 a + 43\right)\cdot 149^{2} + \left(88 a^{2} + 78 a + 74\right)\cdot 149^{3} + \left(30 a^{2} + 66 a + 14\right)\cdot 149^{4} + \left(72 a^{2} + 97 a + 6\right)\cdot 149^{5} + \left(137 a^{2} + 97 a + 11\right)\cdot 149^{6} + \left(115 a^{2} + 49 a + 121\right)\cdot 149^{7} + \left(61 a^{2} + 21 a + 85\right)\cdot 149^{8} + \left(66 a^{2} + 45 a + 27\right)\cdot 149^{9} +O(149^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 139 a^{2} + 61 a + 99 + \left(44 a^{2} + 125 a + 91\right)\cdot 149 + \left(48 a^{2} + 27 a + 64\right)\cdot 149^{2} + \left(147 a^{2} + 140 a + 43\right)\cdot 149^{3} + \left(79 a + 104\right)\cdot 149^{4} + \left(92 a^{2} + 121 a + 45\right)\cdot 149^{5} + \left(146 a^{2} + 12 a + 29\right)\cdot 149^{6} + \left(147 a^{2} + 119 a + 36\right)\cdot 149^{7} + \left(128 a^{2} + 145 a + 71\right)\cdot 149^{8} + \left(42 a^{2} + 124 a + 129\right)\cdot 149^{9} +O(149^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 139 a^{2} + 89 a + 43 + \left(96 a^{2} + 119 a + 109\right)\cdot 149 + \left(16 a^{2} + 31 a + 12\right)\cdot 149^{2} + \left(96 a^{2} + 137 a + 125\right)\cdot 149^{3} + \left(83 a^{2} + 65 a + 21\right)\cdot 149^{4} + \left(98 a^{2} + 25 a + 41\right)\cdot 149^{5} + \left(59 a^{2} + 102 a + 36\right)\cdot 149^{6} + \left(105 a^{2} + 103 a + 139\right)\cdot 149^{7} + \left(107 a^{2} + 41 a + 3\right)\cdot 149^{8} + \left(28 a + 130\right)\cdot 149^{9} +O(149^{10})\)
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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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(2,5)(4,8)$ | $-6$ |
| $9$ | $2$ | $(1,6)(2,5)(3,7)(4,8)$ | $2$ |
| $12$ | $2$ | $(1,3)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $36$ | $2$ | $(1,3)(2,4)$ | $-2$ |
| $36$ | $2$ | $(1,3)(2,5)(4,8)$ | $0$ |
| $16$ | $3$ | $(1,6,7)$ | $0$ |
| $64$ | $3$ | $(1,6,7)(4,5,8)$ | $0$ |
| $12$ | $4$ | $(2,4,5,8)$ | $0$ |
| $36$ | $4$ | $(1,3,6,7)(2,4,5,8)$ | $-2$ |
| $36$ | $4$ | $(1,3,6,7)(2,5)(4,8)$ | $0$ |
| $72$ | $4$ | $(1,2,6,5)(3,4,7,8)$ | $0$ |
| $72$ | $4$ | $(1,3)(2,4,5,8)$ | $2$ |
| $144$ | $4$ | $(1,4,3,2)(5,6)(7,8)$ | $0$ |
| $48$ | $6$ | $(1,7,6)(2,5)(4,8)$ | $0$ |
| $96$ | $6$ | $(1,3)(4,8,5)$ | $0$ |
| $192$ | $6$ | $(1,4,6,5,7,8)(2,3)$ | $0$ |
| $144$ | $8$ | $(1,2,3,4,6,5,7,8)$ | $0$ |
| $96$ | $12$ | $(1,6,7)(2,4,5,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.