Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(321\!\cdots\!000\)\(\medspace = 2^{27} \cdot 3^{22} \cdot 5^{17} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.27993600000.3 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | odd |
| Determinant: | 1.40.2t1.b.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.27993600000.3 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 4x^{6} - 2x^{5} - x^{4} - 14x^{3} + 19x^{2} - 16x - 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 139 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 139 }$:
\( x^{3} + 6x + 137 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 41 + 28\cdot 139 + 124\cdot 139^{2} + 73\cdot 139^{3} + 41\cdot 139^{4} + 49\cdot 139^{5} + 128\cdot 139^{6} + 61\cdot 139^{7} + 135\cdot 139^{8} + 4\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 42 + 88\cdot 139 + 69\cdot 139^{2} + 121\cdot 139^{3} + 25\cdot 139^{4} + 65\cdot 139^{5} + 17\cdot 139^{6} + 10\cdot 139^{7} + 46\cdot 139^{8} + 129\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 a^{2} + 77 a + 133 + \left(134 a^{2} + 107 a + 134\right)\cdot 139 + \left(127 a^{2} + 54 a + 6\right)\cdot 139^{2} + \left(110 a^{2} + 136 a + 115\right)\cdot 139^{3} + \left(113 a^{2} + 72 a + 31\right)\cdot 139^{4} + \left(13 a^{2} + 101 a + 39\right)\cdot 139^{5} + \left(92 a^{2} + 56 a + 29\right)\cdot 139^{6} + \left(112 a^{2} + 107 a + 89\right)\cdot 139^{7} + \left(78 a^{2} + 74 a + 129\right)\cdot 139^{8} + \left(51 a^{2} + 2 a + 87\right)\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 35 a^{2} + 8 a + 54 + \left(108 a^{2} + 105 a + 31\right)\cdot 139 + \left(138 a^{2} + 109 a + 50\right)\cdot 139^{2} + \left(137 a^{2} + 102 a + 84\right)\cdot 139^{3} + \left(136 a^{2} + 34 a + 124\right)\cdot 139^{4} + \left(6 a^{2} + 6 a + 11\right)\cdot 139^{5} + \left(81 a^{2} + 83 a + 124\right)\cdot 139^{6} + \left(48 a^{2} + 3 a + 110\right)\cdot 139^{7} + \left(110 a^{2} + 132 a + 116\right)\cdot 139^{8} + \left(69 a^{2} + 38 a + 21\right)\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 63 a^{2} + 119 a + 80 + \left(8 a^{2} + 119 a + 118\right)\cdot 139 + \left(138 a^{2} + 66 a + 65\right)\cdot 139^{2} + \left(4 a^{2} + 92 a + 5\right)\cdot 139^{3} + \left(110 a^{2} + 46 a + 7\right)\cdot 139^{4} + \left(20 a^{2} + 42 a + 61\right)\cdot 139^{5} + \left(103 a^{2} + 63 a + 54\right)\cdot 139^{6} + \left(47 a^{2} + 134 a + 111\right)\cdot 139^{7} + \left(86 a^{2} + 36 a + 99\right)\cdot 139^{8} + \left(55 a^{2} + 109 a + 110\right)\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 84 a^{2} + 13 a + 25 + \left(37 a^{2} + 80 a + 96\right)\cdot 139 + \left(104 a^{2} + 108 a + 69\right)\cdot 139^{2} + \left(63 a^{2} + 64 a + 101\right)\cdot 139^{3} + \left(16 a^{2} + 41 a + 49\right)\cdot 139^{4} + \left(135 a^{2} + 70 a + 101\right)\cdot 139^{5} + \left(70 a^{2} + 103 a + 64\right)\cdot 139^{6} + \left(2 a^{2} + 49 a + 69\right)\cdot 139^{7} + \left(106 a^{2} + 56 a + 39\right)\cdot 139^{8} + \left(97 a^{2} + 23 a + 1\right)\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 84 a^{2} + 54 a + 111 + \left(35 a^{2} + 65 a + 18\right)\cdot 139 + \left(11 a^{2} + 113 a + 96\right)\cdot 139^{2} + \left(29 a^{2} + 38 a + 65\right)\cdot 139^{3} + \left(27 a^{2} + 31 a + 102\right)\cdot 139^{4} + \left(118 a^{2} + 31 a + 39\right)\cdot 139^{5} + \left(104 a^{2} + 138 a + 80\right)\cdot 139^{6} + \left(116 a^{2} + 27 a + 105\right)\cdot 139^{7} + \left(88 a^{2} + 71 a + 30\right)\cdot 139^{8} + \left(17 a^{2} + 97 a + 91\right)\cdot 139^{9} +O(139^{10})\)
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| $r_{ 8 }$ | $=$ |
\( 131 a^{2} + 7 a + 74 + \left(92 a^{2} + 78 a + 39\right)\cdot 139 + \left(35 a^{2} + 102 a + 73\right)\cdot 139^{2} + \left(70 a^{2} + 120 a + 127\right)\cdot 139^{3} + \left(12 a^{2} + 50 a + 33\right)\cdot 139^{4} + \left(122 a^{2} + 26 a + 49\right)\cdot 139^{5} + \left(103 a^{2} + 111 a + 57\right)\cdot 139^{6} + \left(88 a^{2} + 93 a + 136\right)\cdot 139^{7} + \left(85 a^{2} + 45 a + 96\right)\cdot 139^{8} + \left(124 a^{2} + 6 a + 108\right)\cdot 139^{9} +O(139^{10})\)
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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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(1,4)(3,7)$ | $-6$ |
| $9$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $2$ |
| $12$ | $2$ | $(1,3)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $36$ | $2$ | $(1,3)(2,5)$ | $-2$ |
| $36$ | $2$ | $(1,3)(2,6)(5,8)$ | $0$ |
| $16$ | $3$ | $(3,4,7)$ | $0$ |
| $64$ | $3$ | $(3,4,7)(5,6,8)$ | $0$ |
| $12$ | $4$ | $(1,3,4,7)$ | $0$ |
| $36$ | $4$ | $(1,3,4,7)(2,5,6,8)$ | $-2$ |
| $36$ | $4$ | $(1,4)(2,5,6,8)(3,7)$ | $0$ |
| $72$ | $4$ | $(1,6,4,2)(3,8,7,5)$ | $0$ |
| $72$ | $4$ | $(1,3)(2,5,6,8)$ | $2$ |
| $144$ | $4$ | $(1,5,3,2)(4,6)(7,8)$ | $0$ |
| $48$ | $6$ | $(2,6)(3,7,4)(5,8)$ | $0$ |
| $96$ | $6$ | $(1,3)(5,8,6)$ | $0$ |
| $192$ | $6$ | $(1,2)(3,6,4,8,7,5)$ | $0$ |
| $144$ | $8$ | $(1,5,3,6,4,8,7,2)$ | $0$ |
| $96$ | $12$ | $(2,5,6,8)(3,4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.