Basic invariants
| Dimension: | $8$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(2434731649\)\(\medspace = 7^{4} \cdot 19^{2} \cdot 53^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.1.7148031401.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T213 |
| Parity: | even |
| Projective image: | $S_3\wr S_3$ |
| Projective field: | Galois closure of 9.1.7148031401.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 127 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$:
\( x^{3} + 3x + 124 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 43 a^{2} + 70 a + 115 + \left(105 a^{2} + 80 a + 42\right)\cdot 127 + \left(57 a^{2} + 25 a + 34\right)\cdot 127^{2} + \left(4 a^{2} + 20 a + 7\right)\cdot 127^{3} + \left(55 a^{2} + 19 a + 12\right)\cdot 127^{4} + \left(100 a^{2} + 88 a + 83\right)\cdot 127^{5} + \left(61 a^{2} + 52 a + 3\right)\cdot 127^{6} + \left(118 a^{2} + 31 a + 53\right)\cdot 127^{7} + \left(83 a^{2} + 125 a + 39\right)\cdot 127^{8} + \left(72 a^{2} + 75 a + 113\right)\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 47 a^{2} + 126 a + 16 + \left(53 a^{2} + 91 a + 2\right)\cdot 127 + \left(5 a^{2} + 105 a + 118\right)\cdot 127^{2} + \left(57 a^{2} + 74 a + 122\right)\cdot 127^{3} + \left(85 a^{2} + 115 a + 103\right)\cdot 127^{4} + \left(71 a^{2} + 83 a + 124\right)\cdot 127^{5} + \left(33 a^{2} + 44 a + 46\right)\cdot 127^{6} + \left(2 a^{2} + 85 a + 103\right)\cdot 127^{7} + \left(125 a^{2} + 6 a + 113\right)\cdot 127^{8} + \left(85 a^{2} + 46 a + 66\right)\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 65 a^{2} + 57 a + 53 + \left(70 a^{2} + 89 a + 32\right)\cdot 127 + \left(19 a^{2} + 124 a + 13\right)\cdot 127^{2} + \left(96 a^{2} + 15 a + 58\right)\cdot 127^{3} + \left(58 a^{2} + 42 a + 28\right)\cdot 127^{4} + \left(20 a^{2} + 114 a + 50\right)\cdot 127^{5} + \left(101 a^{2} + 9 a + 88\right)\cdot 127^{6} + \left(115 a^{2} + 62 a + 62\right)\cdot 127^{7} + \left(42 a^{2} + 68 a + 96\right)\cdot 127^{8} + \left(110 a^{2} + 40 a + 103\right)\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 91 a^{2} + 19 a + 105 + \left(101 a^{2} + 67 a + 94\right)\cdot 127 + \left(51 a^{2} + 77\right)\cdot 127^{2} + \left(36 a^{2} + 67 a + 65\right)\cdot 127^{3} + \left(14 a^{2} + 113 a + 66\right)\cdot 127^{4} + \left(101 a^{2} + 51 a + 84\right)\cdot 127^{5} + \left(50 a^{2} + 113 a + 114\right)\cdot 127^{6} + \left(28 a^{2} + 18 a + 14\right)\cdot 127^{7} + \left(72 a^{2} + 122 a + 28\right)\cdot 127^{8} + \left(70 a^{2} + 120 a + 24\right)\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 94 a^{2} + 122 a + 110 + \left(6 a^{2} + 85 a + 35\right)\cdot 127 + \left(108 a^{2} + 48 a + 69\right)\cdot 127^{2} + \left(68 a^{2} + 116 a + 19\right)\cdot 127^{3} + \left(23 a^{2} + a + 107\right)\cdot 127^{4} + \left(34 a^{2} + 92 a + 49\right)\cdot 127^{5} + \left(3 a^{2} + 11 a + 113\right)\cdot 127^{6} + \left(99 a^{2} + 119 a + 42\right)\cdot 127^{7} + \left(97 a^{2} + 54 a + 59\right)\cdot 127^{8} + \left(59 a^{2} + 122 a + 14\right)\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 95 a^{2} + 75 a + 92 + \left(49 a^{2} + 78 a + 58\right)\cdot 127 + \left(126 a^{2} + 80 a + 44\right)\cdot 127^{2} + \left(88 a^{2} + 121 a + 49\right)\cdot 127^{3} + \left(44 a^{2} + 82 a + 118\right)\cdot 127^{4} + \left(72 a^{2} + 47 a + 26\right)\cdot 127^{5} + \left(22 a^{2} + 105 a + 52\right)\cdot 127^{6} + \left(39 a^{2} + 72 a + 21\right)\cdot 127^{7} + \left(113 a^{2} + 3 a + 98\right)\cdot 127^{8} + \left(83 a^{2} + 91 a + 8\right)\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 98 a^{2} + 51 a + 119 + \left(81 a^{2} + 97 a + 54\right)\cdot 127 + \left(55 a^{2} + a + 85\right)\cdot 127^{2} + \left(121 a^{2} + 44 a + 108\right)\cdot 127^{3} + \left(53 a^{2} + 98 a + 18\right)\cdot 127^{4} + \left(5 a^{2} + 87 a + 20\right)\cdot 127^{5} + \left(102 a^{2} + 3 a + 90\right)\cdot 127^{6} + \left(109 a^{2} + 46 a + 50\right)\cdot 127^{7} + \left(11 a^{2} + 63 a + 34\right)\cdot 127^{8} + \left(73 a^{2} + 92 a + 29\right)\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 113 a^{2} + 6 a + 21 + \left(66 a^{2} + 76 a + 29\right)\cdot 127 + \left(13 a^{2} + 99 a + 7\right)\cdot 127^{2} + \left(a^{2} + 62 a + 11\right)\cdot 127^{3} + \left(18 a^{2} + 9 a + 96\right)\cdot 127^{4} + \left(21 a^{2} + 78 a + 23\right)\cdot 127^{5} + \left(90 a^{2} + 70 a + 33\right)\cdot 127^{6} + \left(25 a^{2} + 49 a + 23\right)\cdot 127^{7} + \left(31 a^{2} + 65 a + 53\right)\cdot 127^{8} + \left(108 a^{2} + 85 a + 111\right)\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 116 a^{2} + 109 a + 7 + \left(98 a^{2} + 94 a + 30\right)\cdot 127 + \left(69 a^{2} + 20 a + 58\right)\cdot 127^{2} + \left(33 a^{2} + 112 a + 65\right)\cdot 127^{3} + \left(27 a^{2} + 24 a + 83\right)\cdot 127^{4} + \left(81 a^{2} + 118 a + 44\right)\cdot 127^{5} + \left(42 a^{2} + 95 a + 92\right)\cdot 127^{6} + \left(96 a^{2} + 22 a + 8\right)\cdot 127^{7} + \left(56 a^{2} + 125 a + 112\right)\cdot 127^{8} + \left(97 a^{2} + 86 a + 35\right)\cdot 127^{9} +O(127^{10})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 9 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 9 }$ | Character values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $8$ |
| $9$ | $2$ | $(1,7)$ | $0$ |
| $18$ | $2$ | $(1,2)(4,7)(8,9)$ | $4$ |
| $27$ | $2$ | $(1,7)(2,4)$ | $0$ |
| $27$ | $2$ | $(1,7)(2,4)(3,5)$ | $0$ |
| $54$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $6$ | $3$ | $(3,5,6)$ | $-4$ |
| $8$ | $3$ | $(1,7,8)(2,4,9)(3,5,6)$ | $-1$ |
| $12$ | $3$ | $(1,7,8)(3,5,6)$ | $2$ |
| $72$ | $3$ | $(1,2,3)(4,5,7)(6,8,9)$ | $2$ |
| $54$ | $4$ | $(1,2,7,4)(8,9)$ | $0$ |
| $162$ | $4$ | $(1,5,7,3)(2,4)(6,8)$ | $0$ |
| $36$ | $6$ | $(1,2)(3,5,6)(4,7)(8,9)$ | $-2$ |
| $36$ | $6$ | $(1,3,7,5,8,6)$ | $-2$ |
| $36$ | $6$ | $(1,7)(3,5,6)$ | $0$ |
| $36$ | $6$ | $(1,7)(2,4,9)(3,5,6)$ | $0$ |
| $54$ | $6$ | $(1,7)(2,4)(3,6,5)$ | $0$ |
| $72$ | $6$ | $(1,2,8,9,7,4)(3,5,6)$ | $1$ |
| $108$ | $6$ | $(1,3,7,5,8,6)(2,4)$ | $0$ |
| $216$ | $6$ | $(1,4,5,7,2,3)(6,8,9)$ | $0$ |
| $144$ | $9$ | $(1,2,3,7,4,5,8,9,6)$ | $-1$ |
| $108$ | $12$ | $(1,2,7,4)(3,5,6)(8,9)$ | $0$ |