Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 15.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$: $ x^{3} + x + 145 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 37\cdot 151 + 16\cdot 151^{2} + 42\cdot 151^{4} + 27\cdot 151^{5} + 4\cdot 151^{6} + 129\cdot 151^{7} + 49\cdot 151^{8} + 6\cdot 151^{9} + 96\cdot 151^{10} + 135\cdot 151^{11} + 42\cdot 151^{12} + 80\cdot 151^{13} + 15\cdot 151^{14} +O\left(151^{ 15 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 120 + 2\cdot 151 + 41\cdot 151^{2} + 63\cdot 151^{3} + 125\cdot 151^{4} + 50\cdot 151^{5} + 149\cdot 151^{6} + 3\cdot 151^{7} + 73\cdot 151^{8} + 3\cdot 151^{9} + 51\cdot 151^{10} + 122\cdot 151^{11} + 90\cdot 151^{12} + 113\cdot 151^{13} + 114\cdot 151^{14} +O\left(151^{ 15 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 130 + 130\cdot 151 + 20\cdot 151^{2} + 19\cdot 151^{3} + 133\cdot 151^{4} + 84\cdot 151^{5} + 74\cdot 151^{6} + 100\cdot 151^{7} + 104\cdot 151^{8} + 72\cdot 151^{9} + 115\cdot 151^{10} + 129\cdot 151^{11} + 48\cdot 151^{12} + 69\cdot 151^{13} + 58\cdot 151^{14} +O\left(151^{ 15 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 a^{2} + 89 a + 144 + \left(62 a^{2} + 58 a + 136\right)\cdot 151 + \left(116 a^{2} + 4 a + 140\right)\cdot 151^{2} + \left(98 a^{2} + 141 a + 106\right)\cdot 151^{3} + \left(79 a^{2} + 124 a + 114\right)\cdot 151^{4} + \left(70 a^{2} + 146 a + 72\right)\cdot 151^{5} + \left(147 a^{2} + 111 a + 6\right)\cdot 151^{6} + \left(35 a^{2} + 67 a + 97\right)\cdot 151^{7} + \left(101 a^{2} + 120 a + 87\right)\cdot 151^{8} + \left(73 a^{2} + 119 a + 16\right)\cdot 151^{9} + \left(89 a^{2} + 23 a + 61\right)\cdot 151^{10} + \left(113 a^{2} + 81 a + 44\right)\cdot 151^{11} + \left(129 a^{2} + 51 a + 134\right)\cdot 151^{12} + \left(147 a^{2} + 20 a + 106\right)\cdot 151^{13} + \left(59 a^{2} + 126 a + 44\right)\cdot 151^{14} +O\left(151^{ 15 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 a^{2} + 81 a + 117 + \left(82 a^{2} + 31 a + 49\right)\cdot 151 + \left(14 a^{2} + 49 a + 123\right)\cdot 151^{2} + \left(40 a^{2} + 120 a + 67\right)\cdot 151^{3} + \left(65 a^{2} + 146 a + 4\right)\cdot 151^{4} + \left(22 a^{2} + 35 a + 91\right)\cdot 151^{5} + \left(94 a^{2} + 145 a + 71\right)\cdot 151^{6} + \left(139 a^{2} + 88 a + 65\right)\cdot 151^{7} + \left(48 a^{2} + 32 a + 2\right)\cdot 151^{8} + \left(27 a^{2} + 67 a + 36\right)\cdot 151^{9} + \left(96 a^{2} + 61 a + 15\right)\cdot 151^{10} + \left(35 a^{2} + 121 a + 93\right)\cdot 151^{11} + \left(18 a^{2} + 113 a + 9\right)\cdot 151^{12} + \left(27 a^{2} + 131 a + 127\right)\cdot 151^{13} + \left(74 a^{2} + 105 a + 3\right)\cdot 151^{14} +O\left(151^{ 15 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 54 a^{2} + 33 a + 66 + \left(109 a^{2} + 98 a + 71\right)\cdot 151 + \left(21 a^{2} + 18 a + 126\right)\cdot 151^{2} + \left(132 a^{2} + 150 a + 69\right)\cdot 151^{3} + \left(67 a^{2} + 48 a + 34\right)\cdot 151^{4} + \left(75 a^{2} + 39 a + 121\right)\cdot 151^{5} + \left(81 a^{2} + 6 a + 19\right)\cdot 151^{6} + \left(140 a^{2} + 95 a + 144\right)\cdot 151^{7} + \left(101 a^{2} + 134 a + 122\right)\cdot 151^{8} + \left(98 a^{2} + 147 a + 70\right)\cdot 151^{9} + \left(26 a^{2} + 149 a + 29\right)\cdot 151^{10} + \left(14 a^{2} + 21 a + 12\right)\cdot 151^{11} + \left(135 a^{2} + 115 a + 82\right)\cdot 151^{12} + \left(62 a^{2} + 140 a + 46\right)\cdot 151^{13} + \left(a^{2} + 127 a + 84\right)\cdot 151^{14} +O\left(151^{ 15 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 96 a^{2} + 132 a + \left(6 a^{2} + 60 a + 100\right)\cdot 151 + \left(20 a^{2} + 97 a + 76\right)\cdot 151^{2} + \left(12 a^{2} + 40 a + 99\right)\cdot 151^{3} + \left(6 a^{2} + 30 a + 65\right)\cdot 151^{4} + \left(58 a^{2} + 119 a + 64\right)\cdot 151^{5} + \left(60 a^{2} + 44 a + 99\right)\cdot 151^{6} + \left(126 a^{2} + 145 a + 56\right)\cdot 151^{7} + \left(148 a + 121\right)\cdot 151^{8} + \left(50 a^{2} + 114 a\right)\cdot 151^{9} + \left(116 a^{2} + 65 a + 79\right)\cdot 151^{10} + \left(a^{2} + 99 a + 70\right)\cdot 151^{11} + \left(3 a^{2} + 136 a + 150\right)\cdot 151^{12} + \left(127 a^{2} + 149 a + 92\right)\cdot 151^{13} + \left(16 a^{2} + 69 a + 116\right)\cdot 151^{14} +O\left(151^{ 15 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 124 a^{2} + 13 a + 12 + \left(104 a^{2} + 72 a + 18\right)\cdot 151 + \left(119 a^{2} + 128 a + 91\right)\cdot 151^{2} + \left(100 a^{2} + 67 a + 149\right)\cdot 151^{3} + \left(59 a^{2} + 12 a + 28\right)\cdot 151^{4} + \left(149 a^{2} + 16 a + 120\right)\cdot 151^{5} + \left(61 a^{2} + 21 a + 6\right)\cdot 151^{6} + \left(95 a^{2} + 100 a + 114\right)\cdot 151^{7} + \left(88 a^{2} + 36 a + 63\right)\cdot 151^{8} + \left(2 a^{2} + 44 a + 107\right)\cdot 151^{9} + \left(4 a^{2} + 93 a + 64\right)\cdot 151^{10} + \left(106 a^{2} + 113 a + 73\right)\cdot 151^{11} + \left(109 a^{2} + 123 a + 115\right)\cdot 151^{12} + \left(146 a^{2} + 33 a + 1\right)\cdot 151^{13} + \left(41 a^{2} + 69 a + 61\right)\cdot 151^{14} +O\left(151^{ 15 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 124 a^{2} + 105 a + 12 + \left(87 a^{2} + 131 a + 57\right)\cdot 151 + \left(9 a^{2} + 3 a + 118\right)\cdot 151^{2} + \left(69 a^{2} + 84 a + 27\right)\cdot 151^{3} + \left(23 a^{2} + 89 a + 55\right)\cdot 151^{4} + \left(77 a^{2} + 95 a + 122\right)\cdot 151^{5} + \left(7 a^{2} + 123 a + 20\right)\cdot 151^{6} + \left(66 a^{2} + 106 a + 44\right)\cdot 151^{7} + \left(111 a^{2} + 130 a + 129\right)\cdot 151^{8} + \left(49 a^{2} + 109 a + 138\right)\cdot 151^{9} + \left(120 a^{2} + 58 a + 91\right)\cdot 151^{10} + \left(30 a^{2} + 15 a + 73\right)\cdot 151^{11} + \left(57 a^{2} + 63 a + 80\right)\cdot 151^{12} + \left(92 a^{2} + 127 a + 116\right)\cdot 151^{13} + \left(107 a^{2} + 104 a + 104\right)\cdot 151^{14} +O\left(151^{ 15 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(6,9,8)$ |
| $(1,8,2,6,3,9)$ |
| $(4,7,5)$ |
| $(1,3,2)$ |
| $(1,4)(2,5)(3,7)(6,9,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $9$ | $2$ | $(1,4)(2,5)(3,7)$ | $-1$ |
| $1$ | $3$ | $(1,3,2)(4,7,5)(6,8,9)$ | $3 \zeta_{3}$ |
| $1$ | $3$ | $(1,2,3)(4,5,7)(6,9,8)$ | $-3 \zeta_{3} - 3$ |
| $3$ | $3$ | $(4,7,5)$ | $-2 \zeta_{3} - 1$ |
| $3$ | $3$ | $(4,5,7)$ | $2 \zeta_{3} + 1$ |
| $3$ | $3$ | $(1,2,3)(4,5,7)(6,8,9)$ | $-\zeta_{3} + 1$ |
| $3$ | $3$ | $(1,3,2)(4,7,5)(6,9,8)$ | $\zeta_{3} + 2$ |
| $3$ | $3$ | $(1,2,3)(6,9,8)$ | $\zeta_{3} - 1$ |
| $3$ | $3$ | $(1,3,2)(6,8,9)$ | $-\zeta_{3} - 2$ |
| $6$ | $3$ | $(1,3,2)(4,5,7)$ | $0$ |
| $18$ | $3$ | $(1,4,9)(2,5,8)(3,7,6)$ | $0$ |
| $9$ | $6$ | $(1,4)(2,5)(3,7)(6,9,8)$ | $-1$ |
| $9$ | $6$ | $(1,4)(2,5)(3,7)(6,8,9)$ | $-1$ |
| $9$ | $6$ | $(1,4,3,7,2,5)(6,9,8)$ | $\zeta_{3} + 1$ |
| $9$ | $6$ | $(1,5,2,7,3,4)(6,8,9)$ | $-\zeta_{3}$ |
| $9$ | $6$ | $(1,4,2,5,3,7)(6,9,8)$ | $-\zeta_{3}$ |
| $9$ | $6$ | $(1,7,3,5,2,4)(6,8,9)$ | $\zeta_{3} + 1$ |
| $9$ | $6$ | $(1,8,2,6,3,9)$ | $-\zeta_{3}$ |
| $9$ | $6$ | $(1,9,3,6,2,8)$ | $\zeta_{3} + 1$ |
| $18$ | $9$ | $(1,4,8,3,7,9,2,5,6)$ | $0$ |
| $18$ | $9$ | $(1,8,7,2,6,4,3,9,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
