Basic invariants
| Dimension: | $6$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(116024512\)\(\medspace = 2^{6} \cdot 23^{3} \cdot 149 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.2668563776.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T319 |
| Parity: | odd |
| Determinant: | 1.3427.2t1.a.a |
| Projective image: | $S_3\wr S_3$ |
| Projective stem field: | Galois closure of 9.1.2668563776.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - x^{8} - x^{7} - 4x^{6} + 7x^{4} + 7x^{3} + 5x^{2} + 4x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 127 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$:
\( x^{3} + 3x + 124 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 a^{2} + 49 a + 62 + \left(95 a^{2} + 54 a + 59\right)\cdot 127 + \left(112 a^{2} + 30 a + 76\right)\cdot 127^{2} + \left(117 a^{2} + 79 a + 17\right)\cdot 127^{3} + \left(119 a^{2} + 124 a + 80\right)\cdot 127^{4} + \left(103 a^{2} + 47 a + 15\right)\cdot 127^{5} + \left(58 a^{2} + 96 a + 21\right)\cdot 127^{6} + \left(a^{2} + 76 a + 121\right)\cdot 127^{7} + \left(27 a^{2} + 126 a + 45\right)\cdot 127^{8} + \left(74 a^{2} + 126 a + 18\right)\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 a^{2} + 90 a + 75 + \left(45 a^{2} + 88 a + 95\right)\cdot 127 + \left(77 a^{2} + 99 a + 48\right)\cdot 127^{2} + \left(17 a^{2} + 11 a + 107\right)\cdot 127^{3} + \left(62 a^{2} + 95 a + 118\right)\cdot 127^{4} + \left(69 a^{2} + 92 a + 36\right)\cdot 127^{5} + \left(27 a^{2} + 97 a + 28\right)\cdot 127^{6} + \left(14 a^{2} + 126 a + 4\right)\cdot 127^{7} + \left(4 a^{2} + 84 a + 57\right)\cdot 127^{8} + \left(91 a^{2} + 68 a + 118\right)\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 a^{2} + 33 a + 90 + \left(54 a^{2} + 32 a + 104\right)\cdot 127 + \left(8 a^{2} + 98 a + 121\right)\cdot 127^{2} + \left(109 a^{2} + 93 a + 126\right)\cdot 127^{3} + \left(29 a^{2} + 55 a + 26\right)\cdot 127^{4} + \left(81 a^{2} + 58 a + 97\right)\cdot 127^{5} + \left(46 a^{2} + 11 a + 123\right)\cdot 127^{6} + \left(114 a^{2} + 39 a + 92\right)\cdot 127^{7} + \left(57 a^{2} + 38 a + 107\right)\cdot 127^{8} + \left(23 a^{2} + 23 a + 43\right)\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 30 a^{2} + 92 a + 101 + \left(14 a^{2} + 49 a + 33\right)\cdot 127 + \left(121 a^{2} + 6 a + 9\right)\cdot 127^{2} + \left(72 a^{2} + 125 a + 91\right)\cdot 127^{3} + \left(44 a^{2} + 32 a + 83\right)\cdot 127^{4} + \left(105 a^{2} + 105 a + 108\right)\cdot 127^{5} + \left(71 a^{2} + 82 a + 116\right)\cdot 127^{6} + \left(104 a^{2} + 117 a + 57\right)\cdot 127^{7} + \left(81 a^{2} + 101 a + 85\right)\cdot 127^{8} + \left(42 a^{2} + 108 a + 21\right)\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 57 a^{2} + 120 a + 106 + \left(51 a^{2} + 96 a + 58\right)\cdot 127 + \left(48 a^{2} + 67 a + 55\right)\cdot 127^{2} + \left(62 a^{2} + 120 a + 101\right)\cdot 127^{3} + \left(81 a^{2} + 67 a + 31\right)\cdot 127^{4} + \left(91 a^{2} + 91 a + 54\right)\cdot 127^{5} + \left(78 a^{2} + 73 a + 111\right)\cdot 127^{6} + \left(92 a^{2} + 42 a + 48\right)\cdot 127^{7} + \left(88 a^{2} + 46 a + 94\right)\cdot 127^{8} + \left(118 a^{2} + 12 a + 7\right)\cdot 127^{9} +O(127^{10})\)
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| $r_{ 6 }$ | $=$ |
\( 72 a^{2} + 23 a + 9 + \left(19 a^{2} + 93 a + 122\right)\cdot 127 + \left(a^{2} + 111 a + 87\right)\cdot 127^{2} + \left(29 a^{2} + 71 a + 34\right)\cdot 127^{3} + \left(61 a^{2} + 118\right)\cdot 127^{4} + \left(89 a^{2} + 32 a + 49\right)\cdot 127^{5} + \left(40 a^{2} + 81 a + 35\right)\cdot 127^{6} + \left(109 a^{2} + 81 a + 82\right)\cdot 127^{7} + \left(86 a^{2} + 57 a + 90\right)\cdot 127^{8} + \left(77 a^{2} + a + 52\right)\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 80 a^{2} + 72 a + 74 + \left(67 a^{2} + 115 a + 13\right)\cdot 127 + \left(55 a^{2} + 20 a + 5\right)\cdot 127^{2} + \left(36 a^{2} + 117 a + 18\right)\cdot 127^{3} + \left(20 a^{2} + 125 a + 35\right)\cdot 127^{4} + \left(79 a^{2} + 55 a + 56\right)\cdot 127^{5} + \left(27 a^{2} + 73 a + 28\right)\cdot 127^{6} + \left(8 a^{2} + 9 a + 119\right)\cdot 127^{7} + \left(41 a^{2} + 67 a + 3\right)\cdot 127^{8} + \left(120 a^{2} + 76 a + 50\right)\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 103 a^{2} + 45 a + 4 + \left(104 a^{2} + 40 a + 79\right)\cdot 127 + \left(5 a^{2} + 125 a + 116\right)\cdot 127^{2} + \left(27 a^{2} + 80 a + 89\right)\cdot 127^{3} + \left(104 a^{2} + 73 a + 48\right)\cdot 127^{4} + \left(68 a^{2} + 20 a + 72\right)\cdot 127^{5} + \left(21 a^{2} + 19 a + 73\right)\cdot 127^{6} + \left(11 a^{2} + 11 a + 13\right)\cdot 127^{7} + \left(42 a^{2} + 89 a + 76\right)\cdot 127^{8} + \left(29 a^{2} + 103 a + 55\right)\cdot 127^{9} +O(127^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 125 a^{2} + 111 a + 115 + \left(55 a^{2} + 63 a + 67\right)\cdot 127 + \left(77 a^{2} + 74 a + 113\right)\cdot 127^{2} + \left(35 a^{2} + 61 a + 47\right)\cdot 127^{3} + \left(111 a^{2} + 58 a + 91\right)\cdot 127^{4} + \left(72 a^{2} + 3 a + 16\right)\cdot 127^{5} + \left(7 a^{2} + 99 a + 96\right)\cdot 127^{6} + \left(52 a^{2} + 2 a + 94\right)\cdot 127^{7} + \left(78 a^{2} + 23 a + 73\right)\cdot 127^{8} + \left(57 a^{2} + 113 a + 12\right)\cdot 127^{9} +O(127^{10})\)
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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 value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(5,7)$ | $4$ |
| $18$ | $2$ | $(2,5)(3,7)(8,9)$ | $-2$ |
| $27$ | $2$ | $(1,4)(2,3)(5,7)$ | $0$ |
| $27$ | $2$ | $(1,4)(5,7)$ | $2$ |
| $54$ | $2$ | $(1,5)(2,3)(4,7)(6,8)$ | $0$ |
| $6$ | $3$ | $(1,4,6)$ | $3$ |
| $8$ | $3$ | $(1,6,4)(2,9,3)(5,8,7)$ | $-3$ |
| $12$ | $3$ | $(1,6,4)(5,8,7)$ | $0$ |
| $72$ | $3$ | $(1,5,2)(3,4,7)(6,8,9)$ | $0$ |
| $54$ | $4$ | $(1,7,4,5)(6,8)$ | $-2$ |
| $162$ | $4$ | $(1,7,4,5)(2,3)(6,8)$ | $0$ |
| $36$ | $6$ | $(1,4,6)(2,5)(3,7)(8,9)$ | $1$ |
| $36$ | $6$ | $(1,8,6,7,4,5)$ | $-2$ |
| $36$ | $6$ | $(1,4,6)(5,7)$ | $1$ |
| $36$ | $6$ | $(1,4,6)(2,3,9)(5,7)$ | $-2$ |
| $54$ | $6$ | $(1,6,4)(2,3)(5,7)$ | $-1$ |
| $72$ | $6$ | $(1,4,6)(2,8,9,7,3,5)$ | $1$ |
| $108$ | $6$ | $(1,8,6,7,4,5)(2,3)$ | $0$ |
| $216$ | $6$ | $(1,7,3,4,5,2)(6,8,9)$ | $0$ |
| $144$ | $9$ | $(1,8,9,6,7,3,4,5,2)$ | $0$ |
| $108$ | $12$ | $(1,4,6)(2,7,3,5)(8,9)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.