Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(192\!\cdots\!048\)\(\medspace = 2^{64} \cdot 3^{21} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.13759414272.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.13759414272.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 4x^{6} - 12x^{5} + 24x^{4} - 36x^{3} + 24x^{2} - 12x + 3 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 11 + 24\cdot 139 + 22\cdot 139^{2} + 95\cdot 139^{3} + 100\cdot 139^{4} + 103\cdot 139^{5} + 116\cdot 139^{6} + 139^{7} + 13\cdot 139^{8} + 105\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 39 + 120\cdot 139 + 127\cdot 139^{2} + 101\cdot 139^{3} + 69\cdot 139^{4} + 139^{5} + 87\cdot 139^{6} + 124\cdot 139^{7} + 28\cdot 139^{8} + 111\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 36 a^{2} + 53 a + 39 + \left(57 a^{2} + 10 a + 96\right)\cdot 139 + \left(91 a^{2} + 88 a + 137\right)\cdot 139^{2} + \left(36 a^{2} + 61 a + 19\right)\cdot 139^{3} + \left(104 a^{2} + 126 a + 23\right)\cdot 139^{4} + \left(14 a^{2} + 119 a + 12\right)\cdot 139^{5} + \left(92 a^{2} + 76 a + 15\right)\cdot 139^{6} + \left(135 a^{2} + 94 a + 84\right)\cdot 139^{7} + \left(83 a^{2} + 61 a + 94\right)\cdot 139^{8} + \left(92 a^{2} + 5 a + 101\right)\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 47 a^{2} + 77 a + 83 + \left(52 a^{2} + 114 a + 76\right)\cdot 139 + \left(53 a^{2} + 116 a + 124\right)\cdot 139^{2} + \left(74 a^{2} + 50 a + 31\right)\cdot 139^{3} + \left(20 a^{2} + 18 a + 105\right)\cdot 139^{4} + \left(54 a^{2} + 5 a + 30\right)\cdot 139^{5} + \left(11 a^{2} + 38 a + 109\right)\cdot 139^{6} + \left(24 a^{2} + 50 a + 54\right)\cdot 139^{7} + \left(21 a^{2} + 117 a + 121\right)\cdot 139^{8} + \left(64 a^{2} + 120 a + 126\right)\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 56 a^{2} + 9 a + 119 + \left(29 a^{2} + 14 a + 123\right)\cdot 139 + \left(133 a^{2} + 73 a + 26\right)\cdot 139^{2} + \left(27 a^{2} + 26 a + 124\right)\cdot 139^{3} + \left(14 a^{2} + 133 a + 79\right)\cdot 139^{4} + \left(70 a^{2} + 13 a + 94\right)\cdot 139^{5} + \left(35 a^{2} + 24 a + 66\right)\cdot 139^{6} + \left(118 a^{2} + 133 a + 14\right)\cdot 139^{7} + \left(33 a^{2} + 98 a + 33\right)\cdot 139^{8} + \left(121 a^{2} + 12 a + 77\right)\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 78 a^{2} + 16 a + 31 + \left(30 a^{2} + 129 a + 114\right)\cdot 139 + \left(78 a^{2} + 68 a + 73\right)\cdot 139^{2} + \left(7 a^{2} + 91 a + 137\right)\cdot 139^{3} + \left(112 a^{2} + 115 a + 43\right)\cdot 139^{4} + \left(80 a^{2} + a + 103\right)\cdot 139^{5} + \left(70 a^{2} + 57 a + 11\right)\cdot 139^{6} + \left(48 a^{2} + 106 a + 8\right)\cdot 139^{7} + \left(39 a^{2} + 76 a + 14\right)\cdot 139^{8} + \left(5 a^{2} + 109 a + 125\right)\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 97 a^{2} + 98 a + 107 + \left(29 a^{2} + 106 a + 110\right)\cdot 139 + \left(47 a^{2} + 94 a + 88\right)\cdot 139^{2} + \left(20 a^{2} + 70 a + 49\right)\cdot 139^{3} + \left(28 a^{2} + 7 a + 125\right)\cdot 139^{4} + \left(106 a^{2} + 130 a + 65\right)\cdot 139^{5} + \left(5 a^{2} + 47 a + 30\right)\cdot 139^{6} + \left(65 a^{2} + 36 a + 74\right)\cdot 139^{7} + \left(89 a^{2} + 4 a + 75\right)\cdot 139^{8} + \left(2 a^{2} + 62 a + 114\right)\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 103 a^{2} + 25 a + 131 + \left(78 a^{2} + 42 a + 28\right)\cdot 139 + \left(13 a^{2} + 114 a + 93\right)\cdot 139^{2} + \left(111 a^{2} + 115 a + 134\right)\cdot 139^{3} + \left(137 a^{2} + 15 a + 7\right)\cdot 139^{4} + \left(90 a^{2} + 7 a + 5\right)\cdot 139^{5} + \left(62 a^{2} + 34 a + 119\right)\cdot 139^{6} + \left(25 a^{2} + 135 a + 54\right)\cdot 139^{7} + \left(10 a^{2} + 57 a + 36\right)\cdot 139^{8} + \left(131 a^{2} + 106 a + 72\right)\cdot 139^{9} +O(139^{10})\)
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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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(2,4)(3,5)$ | $-6$ |
| $9$ | $2$ | $(1,7)(2,4)(3,5)(6,8)$ | $2$ |
| $12$ | $2$ | $(2,3)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,6)(4,7)(5,8)$ | $0$ |
| $36$ | $2$ | $(1,6)(2,3)$ | $-2$ |
| $36$ | $2$ | $(1,7)(2,3)(6,8)$ | $0$ |
| $16$ | $3$ | $(6,8,7)$ | $0$ |
| $64$ | $3$ | $(2,4,5)(6,7,8)$ | $0$ |
| $12$ | $4$ | $(2,3,4,5)$ | $0$ |
| $36$ | $4$ | $(1,6,7,8)(2,3,4,5)$ | $-2$ |
| $36$ | $4$ | $(1,6,7,8)(2,4)(3,5)$ | $0$ |
| $72$ | $4$ | $(1,2,7,4)(3,8,5,6)$ | $0$ |
| $72$ | $4$ | $(1,6,7,8)(2,3)$ | $2$ |
| $144$ | $4$ | $(1,2,6,3)(4,7)(5,8)$ | $0$ |
| $48$ | $6$ | $(2,4)(3,5)(6,7,8)$ | $0$ |
| $96$ | $6$ | $(2,3)(6,8,7)$ | $0$ |
| $192$ | $6$ | $(1,3)(2,6,4,7,5,8)$ | $0$ |
| $144$ | $8$ | $(1,2,6,3,7,4,8,5)$ | $0$ |
| $96$ | $12$ | $(2,3,4,5)(6,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.