Basic invariants
| Dimension: | $8$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(40575253489\)\(\medspace = 17^{6} \cdot 41^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.5.5756350841.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T213 |
| Parity: | even |
| Projective image: | $S_3\wr S_3$ |
| Projective field: | Galois closure of 9.5.5756350841.1 |
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:
| $r_{ 1 }$ | $=$ |
\( 22 + 45\cdot 127 + 97\cdot 127^{2} + 39\cdot 127^{3} + 18\cdot 127^{4} + 24\cdot 127^{5} + 25\cdot 127^{6} + 19\cdot 127^{7} + 64\cdot 127^{8} + 99\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 66 + 41\cdot 127 + 91\cdot 127^{2} + 124\cdot 127^{3} + 27\cdot 127^{4} + 2\cdot 127^{5} + 81\cdot 127^{6} + 86\cdot 127^{7} + 96\cdot 127^{8} + 17\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 110 + 19\cdot 127 + 124\cdot 127^{2} + 70\cdot 127^{3} + 51\cdot 127^{4} + 31\cdot 127^{6} + 64\cdot 127^{7} + 121\cdot 127^{8} + 87\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 79 a + 83 + \left(14 a^{2} + 125 a + 74\right)\cdot 127 + \left(101 a^{2} + 19 a + 124\right)\cdot 127^{2} + \left(15 a^{2} + 39 a + 33\right)\cdot 127^{3} + \left(37 a^{2} + 94 a + 10\right)\cdot 127^{4} + \left(105 a^{2} + 94 a + 106\right)\cdot 127^{5} + \left(87 a^{2} + 64 a + 60\right)\cdot 127^{6} + \left(105 a^{2} + 4 a + 62\right)\cdot 127^{7} + \left(41 a^{2} + 20 a + 24\right)\cdot 127^{8} + \left(97 a^{2} + 84 a + 59\right)\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 a^{2} + 35 a + 43 + \left(93 a^{2} + 42 a + 19\right)\cdot 127 + \left(118 a^{2} + 62 a + 126\right)\cdot 127^{2} + \left(122 a^{2} + 15 a + 37\right)\cdot 127^{3} + \left(124 a^{2} + 63 a + 27\right)\cdot 127^{4} + \left(77 a^{2} + 118 a + 82\right)\cdot 127^{5} + \left(44 a^{2} + 90 a + 73\right)\cdot 127^{6} + \left(34 a^{2} + 47 a + 118\right)\cdot 127^{7} + \left(34 a^{2} + 92 a + 75\right)\cdot 127^{8} + \left(8 a^{2} + 24 a + 83\right)\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 40 a^{2} + 46 a + 101 + \left(36 a^{2} + 123 a + 32\right)\cdot 127 + \left(125 a^{2} + 103 a + 12\right)\cdot 127^{2} + \left(110 a^{2} + 9 a + 14\right)\cdot 127^{3} + \left(3 a^{2} + 111 a + 39\right)\cdot 127^{4} + \left(77 a^{2} + 11 a + 80\right)\cdot 127^{5} + \left(25 a^{2} + 45 a + 35\right)\cdot 127^{6} + \left(45 a^{2} + 25 a + 13\right)\cdot 127^{7} + \left(74 a^{2} + 62 a + 29\right)\cdot 127^{8} + \left(40 a^{2} + 40 a + 21\right)\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 57 a^{2} + 46 a + 70 + \left(79 a^{2} + 107 a + 78\right)\cdot 127 + \left(8 a^{2} + 57 a + 66\right)\cdot 127^{2} + \left(30 a^{2} + 72 a + 62\right)\cdot 127^{3} + \left(51 a^{2} + 10 a + 38\right)\cdot 127^{4} + \left(67 a^{2} + 124 a + 30\right)\cdot 127^{5} + \left(61 a^{2} + 101 a + 8\right)\cdot 127^{6} + \left(31 a^{2} + 28 a + 41\right)\cdot 127^{7} + \left(92 a^{2} + 108 a + 125\right)\cdot 127^{8} + \left(26 a^{2} + 16 a + 44\right)\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 70 a^{2} + 2 a + 96 + \left(33 a^{2} + 21 a + 113\right)\cdot 127 + \left(17 a^{2} + 49 a + 83\right)\cdot 127^{2} + \left(81 a^{2} + 15 a + 37\right)\cdot 127^{3} + \left(38 a^{2} + 22 a + 13\right)\cdot 127^{4} + \left(81 a^{2} + 35 a + 58\right)\cdot 127^{5} + \left(104 a^{2} + 87 a + 94\right)\cdot 127^{6} + \left(116 a^{2} + 93 a + 84\right)\cdot 127^{7} + \left(119 a^{2} + 125 a + 53\right)\cdot 127^{8} + \left(2 a^{2} + 25 a + 124\right)\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 76 a^{2} + 46 a + 46 + \left(124 a^{2} + 88 a + 82\right)\cdot 127 + \left(9 a^{2} + 87 a + 35\right)\cdot 127^{2} + \left(20 a^{2} + 101 a + 86\right)\cdot 127^{3} + \left(125 a^{2} + 79 a + 27\right)\cdot 127^{4} + \left(98 a^{2} + 123 a + 124\right)\cdot 127^{5} + \left(56 a^{2} + 117 a + 97\right)\cdot 127^{6} + \left(47 a^{2} + 53 a + 17\right)\cdot 127^{7} + \left(18 a^{2} + 99 a + 44\right)\cdot 127^{8} + \left(78 a^{2} + 61 a + 96\right)\cdot 127^{9} +O(127^{10})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $8$ |
| $9$ | $2$ | $(1,2)$ | $0$ |
| $18$ | $2$ | $(1,4)(2,7)(3,8)$ | $4$ |
| $27$ | $2$ | $(1,2)(4,7)(5,6)$ | $0$ |
| $27$ | $2$ | $(1,2)(5,6)$ | $0$ |
| $54$ | $2$ | $(1,5)(2,6)(3,9)(4,7)$ | $0$ |
| $6$ | $3$ | $(5,6,9)$ | $-4$ |
| $8$ | $3$ | $(1,3,2)(4,8,7)(5,9,6)$ | $-1$ |
| $12$ | $3$ | $(1,3,2)(5,9,6)$ | $2$ |
| $72$ | $3$ | $(1,4,5)(2,7,6)(3,8,9)$ | $2$ |
| $54$ | $4$ | $(1,5,2,6)(3,9)$ | $0$ |
| $162$ | $4$ | $(1,5,2,6)(3,9)(4,7)$ | $0$ |
| $36$ | $6$ | $(1,4)(2,7)(3,8)(5,6,9)$ | $-2$ |
| $36$ | $6$ | $(1,5,3,9,2,6)$ | $-2$ |
| $36$ | $6$ | $(1,2)(5,6,9)$ | $0$ |
| $36$ | $6$ | $(1,2)(4,7,8)(5,6,9)$ | $0$ |
| $54$ | $6$ | $(1,2)(4,7)(5,9,6)$ | $0$ |
| $72$ | $6$ | $(1,4,3,8,2,7)(5,6,9)$ | $1$ |
| $108$ | $6$ | $(1,5,3,9,2,6)(4,7)$ | $0$ |
| $216$ | $6$ | $(1,4,5,2,7,6)(3,8,9)$ | $0$ |
| $144$ | $9$ | $(1,4,5,3,8,9,2,7,6)$ | $-1$ |
| $108$ | $12$ | $(1,4,2,7)(3,8)(5,6,9)$ | $0$ |