Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(641\!\cdots\!927\)\(\medspace = 3^{38} \cdot 7^{15} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.2977309629.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | odd |
| Determinant: | 1.7.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.0.2977309629.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 8x^{5} - 3x^{4} - 33x^{3} + 79x^{2} - 81x + 39 \)
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The roots of $f$ are computed in an extension of $\Q_{ 157 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 157 }$:
\( x^{2} + 152x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 34 + 33\cdot 157 + 157^{2} + 20\cdot 157^{3} + 101\cdot 157^{4} + 94\cdot 157^{5} + 24\cdot 157^{6} + 144\cdot 157^{7} + 149\cdot 157^{8} + 82\cdot 157^{9} +O(157^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 110 a + 21 + \left(8 a + 19\right)\cdot 157 + \left(92 a + 47\right)\cdot 157^{2} + \left(85 a + 28\right)\cdot 157^{3} + \left(37 a + 147\right)\cdot 157^{4} + \left(47 a + 151\right)\cdot 157^{5} + \left(46 a + 91\right)\cdot 157^{6} + \left(135 a + 103\right)\cdot 157^{7} + \left(16 a + 146\right)\cdot 157^{8} + \left(153 a + 16\right)\cdot 157^{9} +O(157^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 87 a + 100 + \left(91 a + 10\right)\cdot 157 + \left(72 a + 35\right)\cdot 157^{2} + \left(39 a + 32\right)\cdot 157^{3} + \left(106 a + 34\right)\cdot 157^{4} + \left(85 a + 118\right)\cdot 157^{5} + \left(75 a + 8\right)\cdot 157^{6} + \left(85 a + 18\right)\cdot 157^{7} + \left(97 a + 4\right)\cdot 157^{8} + \left(129 a + 7\right)\cdot 157^{9} +O(157^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 105 + 89\cdot 157 + 67\cdot 157^{2} + 66\cdot 157^{3} + 93\cdot 157^{4} + 29\cdot 157^{5} + 133\cdot 157^{6} + 57\cdot 157^{7} + 9\cdot 157^{8} + 104\cdot 157^{9} +O(157^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 87 a + 43 + \left(112 a + 68\right)\cdot 157 + \left(96 a + 121\right)\cdot 157^{2} + \left(19 a + 57\right)\cdot 157^{3} + \left(100 a + 47\right)\cdot 157^{4} + \left(135 a + 60\right)\cdot 157^{5} + \left(51 a + 148\right)\cdot 157^{6} + \left(128 a + 11\right)\cdot 157^{7} + \left(17 a + 6\right)\cdot 157^{8} + \left(70 a + 87\right)\cdot 157^{9} +O(157^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 70 a + 7 + \left(44 a + 73\right)\cdot 157 + \left(60 a + 21\right)\cdot 157^{2} + \left(137 a + 59\right)\cdot 157^{3} + \left(56 a + 57\right)\cdot 157^{4} + \left(21 a + 10\right)\cdot 157^{5} + \left(105 a + 115\right)\cdot 157^{6} + \left(28 a + 130\right)\cdot 157^{7} + \left(139 a + 123\right)\cdot 157^{8} + \left(86 a + 105\right)\cdot 157^{9} +O(157^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 47 a + 100 + \left(148 a + 109\right)\cdot 157 + \left(64 a + 27\right)\cdot 157^{2} + \left(71 a + 50\right)\cdot 157^{3} + \left(119 a + 92\right)\cdot 157^{4} + \left(109 a + 36\right)\cdot 157^{5} + \left(110 a + 119\right)\cdot 157^{6} + \left(21 a + 105\right)\cdot 157^{7} + \left(140 a + 95\right)\cdot 157^{8} + \left(3 a + 137\right)\cdot 157^{9} +O(157^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 70 a + 64 + \left(65 a + 67\right)\cdot 157 + \left(84 a + 149\right)\cdot 157^{2} + \left(117 a + 156\right)\cdot 157^{3} + \left(50 a + 54\right)\cdot 157^{4} + \left(71 a + 126\right)\cdot 157^{5} + \left(81 a + 143\right)\cdot 157^{6} + \left(71 a + 55\right)\cdot 157^{7} + \left(59 a + 92\right)\cdot 157^{8} + \left(27 a + 86\right)\cdot 157^{9} +O(157^{10})\)
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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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(1,4)(3,8)$ | $-6$ |
| $9$ | $2$ | $(1,4)(2,6)(3,8)(5,7)$ | $2$ |
| $12$ | $2$ | $(2,5)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $36$ | $2$ | $(1,3)(2,5)$ | $-2$ |
| $36$ | $2$ | $(1,4)(2,5)(3,8)$ | $0$ |
| $16$ | $3$ | $(2,6,7)$ | $0$ |
| $64$ | $3$ | $(2,6,7)(3,4,8)$ | $0$ |
| $12$ | $4$ | $(1,3,4,8)$ | $0$ |
| $36$ | $4$ | $(1,3,4,8)(2,5,6,7)$ | $-2$ |
| $36$ | $4$ | $(1,4)(2,5,6,7)(3,8)$ | $0$ |
| $72$ | $4$ | $(1,6,4,2)(3,7,8,5)$ | $0$ |
| $72$ | $4$ | $(1,3,4,8)(2,5)$ | $2$ |
| $144$ | $4$ | $(1,2,3,5)(4,6)(7,8)$ | $0$ |
| $48$ | $6$ | $(1,4)(2,7,6)(3,8)$ | $0$ |
| $96$ | $6$ | $(2,5)(3,8,4)$ | $0$ |
| $192$ | $6$ | $(1,5)(2,3,6,4,7,8)$ | $0$ |
| $144$ | $8$ | $(1,5,3,6,4,7,8,2)$ | $0$ |
| $96$ | $12$ | $(1,3,4,8)(2,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.