Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(108\!\cdots\!024\)\(\medspace = 2^{33} \cdot 7^{9} \cdot 11^{12} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.6.82278203392.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | odd |
| Determinant: | 1.56.2t1.b.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.6.82278203392.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 12x^{5} - 30x^{4} + 32x^{3} + 28x^{2} - 12x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 137 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 137 }$:
\( x^{3} + 6x + 134 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 64 + 58\cdot 137 + 39\cdot 137^{2} + 56\cdot 137^{4} + 45\cdot 137^{5} + 66\cdot 137^{6} + 131\cdot 137^{7} + 15\cdot 137^{8} + 65\cdot 137^{9} +O(137^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 86 + 132\cdot 137 + 17\cdot 137^{2} + 35\cdot 137^{3} + 108\cdot 137^{4} + 91\cdot 137^{5} + 97\cdot 137^{6} + 58\cdot 137^{7} + 102\cdot 137^{8} + 118\cdot 137^{9} +O(137^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 a^{2} + 79 a + 126 + \left(33 a^{2} + 49 a + 112\right)\cdot 137 + \left(40 a^{2} + 98 a + 10\right)\cdot 137^{2} + \left(98 a^{2} + 88 a + 119\right)\cdot 137^{3} + \left(3 a^{2} + 24 a + 41\right)\cdot 137^{4} + \left(107 a^{2} + 129 a + 93\right)\cdot 137^{5} + \left(120 a^{2} + 15 a + 95\right)\cdot 137^{6} + \left(14 a^{2} + 3 a + 15\right)\cdot 137^{7} + \left(43 a^{2} + 42 a + 30\right)\cdot 137^{8} + \left(107 a^{2} + 6 a + 42\right)\cdot 137^{9} +O(137^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 56 a^{2} + 9 a + 104 + \left(25 a^{2} + 8 a + 11\right)\cdot 137 + \left(37 a^{2} + 32 a + 97\right)\cdot 137^{2} + \left(72 a^{2} + 57 a + 94\right)\cdot 137^{3} + \left(93 a^{2} + 45 a + 109\right)\cdot 137^{4} + \left(29 a^{2} + 76 a + 133\right)\cdot 137^{5} + \left(69 a^{2} + 116 a + 15\right)\cdot 137^{6} + \left(81 a^{2} + 17 a + 78\right)\cdot 137^{7} + \left(99 a^{2} + 33 a + 44\right)\cdot 137^{8} + \left(65 a^{2} + 100 a + 86\right)\cdot 137^{9} +O(137^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 93 a^{2} + 51 a + 115 + \left(104 a^{2} + 133 a + 54\right)\cdot 137 + \left(104 a^{2} + 93 a + 93\right)\cdot 137^{2} + \left(20 a^{2} + 33 a + 25\right)\cdot 137^{3} + \left(50 a^{2} + 54 a + 73\right)\cdot 137^{4} + \left(107 a^{2} + 50 a + 33\right)\cdot 137^{5} + \left(19 a^{2} + 71 a + 92\right)\cdot 137^{6} + \left(a^{2} + 15 a + 30\right)\cdot 137^{7} + \left(79 a^{2} + 29 a + 99\right)\cdot 137^{8} + \left(5 a^{2} + 39 a + 119\right)\cdot 137^{9} +O(137^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 125 a^{2} + 77 a + 106 + \left(6 a^{2} + 132 a + 74\right)\cdot 137 + \left(132 a^{2} + 10 a + 65\right)\cdot 137^{2} + \left(43 a^{2} + 46 a + 118\right)\cdot 137^{3} + \left(130 a^{2} + 37 a + 119\right)\cdot 137^{4} + \left(136 a^{2} + 10 a + 14\right)\cdot 137^{5} + \left(47 a^{2} + 86 a + 68\right)\cdot 137^{6} + \left(54 a^{2} + 103 a + 106\right)\cdot 137^{7} + \left(95 a^{2} + 74 a + 27\right)\cdot 137^{8} + \left(65 a^{2} + 134 a + 86\right)\cdot 137^{9} +O(137^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 130 a^{2} + 63 a + 42 + \left(73 a^{2} + 133 a + 2\right)\cdot 137 + \left(69 a^{2} + 44 a + 128\right)\cdot 137^{2} + \left(6 a^{2} + 27 a + 25\right)\cdot 137^{3} + \left(17 a^{2} + 15 a + 95\right)\cdot 137^{4} + \left(105 a^{2} + 111 a + 85\right)\cdot 137^{5} + \left(112 a^{2} + 124 a + 63\right)\cdot 137^{6} + \left(104 a^{2} + 49 a + 101\right)\cdot 137^{7} + \left(27 a^{2} + 41 a + 105\right)\cdot 137^{8} + \left(41 a^{2} + 51\right)\cdot 137^{9} +O(137^{10})\)
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| $r_{ 8 }$ | $=$ |
\( 130 a^{2} + 132 a + 42 + \left(29 a^{2} + 90 a + 100\right)\cdot 137 + \left(27 a^{2} + 130 a + 95\right)\cdot 137^{2} + \left(32 a^{2} + 20 a + 128\right)\cdot 137^{3} + \left(116 a^{2} + 97 a + 80\right)\cdot 137^{4} + \left(61 a^{2} + 33 a + 49\right)\cdot 137^{5} + \left(40 a^{2} + 133 a + 48\right)\cdot 137^{6} + \left(17 a^{2} + 83 a + 25\right)\cdot 137^{7} + \left(66 a^{2} + 53 a + 122\right)\cdot 137^{8} + \left(125 a^{2} + 130 a + 114\right)\cdot 137^{9} +O(137^{10})\)
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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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(1,7)(3,8)$ | $-6$ |
| $9$ | $2$ | $(1,7)(2,5)(3,8)(4,6)$ | $2$ |
| $12$ | $2$ | $(1,3)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $0$ |
| $36$ | $2$ | $(1,3)(2,4)$ | $-2$ |
| $36$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
| $16$ | $3$ | $(3,7,8)$ | $0$ |
| $64$ | $3$ | $(3,7,8)(4,5,6)$ | $0$ |
| $12$ | $4$ | $(1,3,7,8)$ | $0$ |
| $36$ | $4$ | $(1,3,7,8)(2,4,5,6)$ | $-2$ |
| $36$ | $4$ | $(1,7)(2,4,5,6)(3,8)$ | $0$ |
| $72$ | $4$ | $(1,5,7,2)(3,6,8,4)$ | $0$ |
| $72$ | $4$ | $(1,3)(2,4,5,6)$ | $2$ |
| $144$ | $4$ | $(1,4,3,2)(5,7)(6,8)$ | $0$ |
| $48$ | $6$ | $(2,5)(3,8,7)(4,6)$ | $0$ |
| $96$ | $6$ | $(1,3)(4,6,5)$ | $0$ |
| $192$ | $6$ | $(1,2)(3,5,7,6,8,4)$ | $0$ |
| $144$ | $8$ | $(1,4,3,5,7,6,8,2)$ | $0$ |
| $96$ | $12$ | $(2,4,5,6)(3,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.