Basic invariants
| Dimension: | $8$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(9120059001\)\(\medspace = 3^{12} \cdot 131^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.32257648686537.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T213 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_3\wr S_3$ |
| Projective stem field: | Galois closure of 9.1.32257648686537.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{8} + 9x^{7} + 21x^{6} - 6x^{5} + 72x^{4} + 33x^{3} + 60x^{2} + 64 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$:
\( x^{3} + x + 145 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 a^{2} + 8 a + 134 + \left(54 a^{2} + 122 a + 140\right)\cdot 151 + \left(139 a^{2} + 52 a + 54\right)\cdot 151^{2} + \left(38 a^{2} + 12 a + 86\right)\cdot 151^{3} + \left(59 a^{2} + 45 a + 4\right)\cdot 151^{4} + \left(3 a^{2} + 109 a + 1\right)\cdot 151^{5} + \left(143 a^{2} + 16 a + 25\right)\cdot 151^{6} + \left(96 a^{2} + 19 a + 77\right)\cdot 151^{7} + \left(68 a^{2} + 97 a + 129\right)\cdot 151^{8} + \left(19 a^{2} + 93 a + 27\right)\cdot 151^{9} +O(151^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 a^{2} + 38 a + 77 + \left(94 a^{2} + 73 a + 89\right)\cdot 151 + \left(38 a^{2} + 42 a + 40\right)\cdot 151^{2} + \left(66 a^{2} + 52 a + 44\right)\cdot 151^{3} + \left(74 a^{2} + 130 a + 102\right)\cdot 151^{4} + \left(132 a^{2} + 134 a + 124\right)\cdot 151^{5} + \left(8 a^{2} + 113 a + 6\right)\cdot 151^{6} + \left(11 a^{2} + 130 a + 21\right)\cdot 151^{7} + \left(41 a^{2} + 109 a + 51\right)\cdot 151^{8} + \left(133 a^{2} + 93 a + 21\right)\cdot 151^{9} +O(151^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a^{2} + 105 a + 89 + \left(8 a^{2} + 109 a + 9\right)\cdot 151 + \left(80 a^{2} + 42 a + 116\right)\cdot 151^{2} + \left(28 a^{2} + 39 a + 129\right)\cdot 151^{3} + \left(130 a^{2} + 123 a + 51\right)\cdot 151^{4} + \left(22 a^{2} + 142 a + 64\right)\cdot 151^{5} + \left(127 a^{2} + 37 a + 14\right)\cdot 151^{6} + \left(82 a^{2} + 110 a + 118\right)\cdot 151^{7} + \left(38 a^{2} + 77 a + 8\right)\cdot 151^{8} + \left(4 a^{2} + 100 a + 68\right)\cdot 151^{9} +O(151^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 28 a^{2} + 116 a + 77 + \left(63 a^{2} + 7 a + 61\right)\cdot 151 + \left(49 a^{2} + 48 a + 106\right)\cdot 151^{2} + \left(29 a^{2} + 88 a + 59\right)\cdot 151^{3} + \left(2 a^{2} + 18 a + 84\right)\cdot 151^{4} + \left(2 a^{2} + 121 a + 16\right)\cdot 151^{5} + \left(144 a^{2} + 28 a + 115\right)\cdot 151^{6} + \left(67 a^{2} + 58 a + 119\right)\cdot 151^{7} + \left(102 a^{2} + 130 a + 111\right)\cdot 151^{8} + \left(61 a^{2} + 46 a + 93\right)\cdot 151^{9} +O(151^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 50 a^{2} + 117 a + 50 + \left(104 a^{2} + 38 a + 96\right)\cdot 151 + \left(56 a^{2} + 31 a + 52\right)\cdot 151^{2} + \left(131 a^{2} + 131 a + 37\right)\cdot 151^{3} + \left(102 a^{2} + 126 a + 121\right)\cdot 151^{4} + \left(108 a^{2} + 21 a + 108\right)\cdot 151^{5} + \left(11 a^{2} + 118 a + 8\right)\cdot 151^{6} + \left(39 a^{2} + 143 a + 90\right)\cdot 151^{7} + \left(111 a^{2} + 128 a + 47\right)\cdot 151^{8} + \left(88 a^{2} + 150 a + 92\right)\cdot 151^{9} +O(151^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 86 a^{2} + 147 a + 74 + \left(103 a^{2} + 38 a + 45\right)\cdot 151 + \left(55 a^{2} + 77 a + 102\right)\cdot 151^{2} + \left(104 a^{2} + 118 a + 69\right)\cdot 151^{3} + \left(124 a^{2} + 44 a + 85\right)\cdot 151^{4} + \left(60 a^{2} + 145 a + 26\right)\cdot 151^{5} + \left(130 a^{2} + 69 a + 138\right)\cdot 151^{6} + \left(100 a^{2} + 27 a + 80\right)\cdot 151^{7} + \left(149 a^{2} + 63 a + 123\right)\cdot 151^{8} + \left(79 a^{2} + 57 a + 136\right)\cdot 151^{9} +O(151^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 123 a^{2} + 38 a + 8 + \left(88 a^{2} + 70 a + 13\right)\cdot 151 + \left(82 a^{2} + 55 a + 17\right)\cdot 151^{2} + \left(83 a^{2} + 99 a + 116\right)\cdot 151^{3} + \left(112 a^{2} + 133 a + 140\right)\cdot 151^{4} + \left(124 a^{2} + 49 a + 81\right)\cdot 151^{5} + \left(31 a^{2} + 96 a + 51\right)\cdot 151^{6} + \left(122 a^{2} + 21 a + 144\right)\cdot 151^{7} + \left(43 a^{2} + 127 a + 112\right)\cdot 151^{8} + \left(127 a^{2} + 107 a + 99\right)\cdot 151^{9} +O(151^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 132 a^{2} + 111 a + 96 + \left(113 a^{2} + 13 a + 145\right)\cdot 151 + \left(48 a^{2} + 42 a + 105\right)\cdot 151^{2} + \left(117 a^{2} + 98 a + 17\right)\cdot 151^{3} + \left(71 a^{2} + 77 a + 30\right)\cdot 151^{4} + \left(148 a^{2} + 14 a + 114\right)\cdot 151^{5} + \left(81 a^{2} + 70 a + 73\right)\cdot 151^{6} + \left(137 a^{2} + 52 a + 65\right)\cdot 151^{7} + \left(2 a^{2} + 79 a + 45\right)\cdot 151^{8} + \left(34 a^{2} + 91 a + 75\right)\cdot 151^{9} +O(151^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 142 a^{2} + 75 a + 2 + \left(124 a^{2} + 129 a + 2\right)\cdot 151 + \left(52 a^{2} + 60 a + 8\right)\cdot 151^{2} + \left(4 a^{2} + 115 a + 43\right)\cdot 151^{3} + \left(77 a^{2} + 54 a + 134\right)\cdot 151^{4} + \left(15 a + 65\right)\cdot 151^{5} + \left(76 a^{2} + 52 a + 19\right)\cdot 151^{6} + \left(96 a^{2} + 40 a + 38\right)\cdot 151^{7} + \left(45 a^{2} + 92 a + 124\right)\cdot 151^{8} + \left(55 a^{2} + 12 a + 139\right)\cdot 151^{9} +O(151^{10})\)
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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 |
| $1$ | $1$ | $()$ | $8$ |
| $9$ | $2$ | $(1,5)$ | $0$ |
| $18$ | $2$ | $(1,2)(3,5)(4,8)$ | $4$ |
| $27$ | $2$ | $(1,5)(2,3)(6,7)$ | $0$ |
| $27$ | $2$ | $(1,5)(6,7)$ | $0$ |
| $54$ | $2$ | $(1,5)(2,6)(3,7)(4,9)$ | $0$ |
| $6$ | $3$ | $(6,7,9)$ | $-4$ |
| $8$ | $3$ | $(1,8,5)(2,4,3)(6,9,7)$ | $-1$ |
| $12$ | $3$ | $(2,3,4)(6,7,9)$ | $2$ |
| $72$ | $3$ | $(1,2,6)(3,7,5)(4,9,8)$ | $2$ |
| $54$ | $4$ | $(1,6,5,7)(8,9)$ | $0$ |
| $162$ | $4$ | $(1,6,5,7)(3,4)(8,9)$ | $0$ |
| $36$ | $6$ | $(1,2)(3,5)(4,8)(6,7,9)$ | $-2$ |
| $36$ | $6$ | $(1,6,8,9,5,7)$ | $-2$ |
| $36$ | $6$ | $(1,5)(6,7,9)$ | $0$ |
| $36$ | $6$ | $(1,5)(2,3,4)(6,7,9)$ | $0$ |
| $54$ | $6$ | $(1,5)(2,4,3)(6,7)$ | $0$ |
| $72$ | $6$ | $(1,2,5,3,8,4)(6,7,9)$ | $1$ |
| $108$ | $6$ | $(1,5)(2,6,3,7,4,9)$ | $0$ |
| $216$ | $6$ | $(1,2,6,5,3,7)(4,9,8)$ | $0$ |
| $144$ | $9$ | $(1,2,6,8,4,9,5,3,7)$ | $-1$ |
| $108$ | $12$ | $(1,6,5,7)(2,3,4)(8,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.