Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$: $ x^{3} + x + 145 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 84\cdot 151 + 126\cdot 151^{2} + 33\cdot 151^{4} + 10\cdot 151^{5} + 66\cdot 151^{7} + 32\cdot 151^{8} + 38\cdot 151^{9} + 112\cdot 151^{10} +O\left(151^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 58 + 28\cdot 151 + 135\cdot 151^{2} + 57\cdot 151^{3} + 55\cdot 151^{4} + 21\cdot 151^{5} + 36\cdot 151^{6} + 142\cdot 151^{7} + 19\cdot 151^{8} + 129\cdot 151^{9} + 104\cdot 151^{10} +O\left(151^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 88 + 38\cdot 151 + 40\cdot 151^{2} + 92\cdot 151^{3} + 62\cdot 151^{4} + 119\cdot 151^{5} + 114\cdot 151^{6} + 93\cdot 151^{7} + 98\cdot 151^{8} + 134\cdot 151^{9} + 84\cdot 151^{10} +O\left(151^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 61 a^{2} + 106 a + 91 + \left(113 a^{2} + 85 a + 75\right)\cdot 151 + \left(107 a^{2} + 101 a + 21\right)\cdot 151^{2} + \left(83 a^{2} + 22 a + 106\right)\cdot 151^{3} + \left(100 a^{2} + 116 a + 16\right)\cdot 151^{4} + \left(99 a^{2} + 47 a + 16\right)\cdot 151^{5} + \left(9 a^{2} + 14 a + 107\right)\cdot 151^{6} + \left(147 a^{2} + 34 a + 47\right)\cdot 151^{7} + \left(117 a^{2} + 93 a + 28\right)\cdot 151^{8} + \left(9 a^{2} + 71 a + 107\right)\cdot 151^{9} + \left(5 a^{2} + 51 a + 53\right)\cdot 151^{10} +O\left(151^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 80 a^{2} + 47 a + 3 + \left(127 a^{2} + 139 a + 85\right)\cdot 151 + \left(59 a^{2} + 4 a + 140\right)\cdot 151^{2} + \left(6 a^{2} + 59 a + 104\right)\cdot 151^{3} + \left(25 a^{2} + 16\right)\cdot 151^{4} + \left(13 a^{2} + 134 a + 59\right)\cdot 151^{5} + \left(142 a^{2} + 98 a + 44\right)\cdot 151^{6} + \left(144 a^{2} + 29 a + 46\right)\cdot 151^{7} + \left(35 a^{2} + 124 a + 74\right)\cdot 151^{8} + \left(87 a^{2} + 115 a + 108\right)\cdot 151^{9} + \left(10 a^{2} + 90 a + 107\right)\cdot 151^{10} +O\left(151^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 88 a^{2} + 118 a + 109 + \left(117 a^{2} + 72 a + 128\right)\cdot 151 + \left(32 a^{2} + 87 a + 21\right)\cdot 151^{2} + \left(142 a^{2} + 131 a + 145\right)\cdot 151^{3} + \left(136 a^{2} + 55 a + 40\right)\cdot 151^{4} + \left(81 a^{2} + 83 a + 4\right)\cdot 151^{5} + \left(91 a^{2} + 84 a + 61\right)\cdot 151^{6} + \left(131 a^{2} + 42 a + 37\right)\cdot 151^{7} + \left(98 a^{2} + 118 a + 116\right)\cdot 151^{8} + \left(62 a^{2} + 139 a + 41\right)\cdot 151^{9} + \left(57 a^{2} + 46 a + 38\right)\cdot 151^{10} +O\left(151^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 92 a^{2} + 92 a + 11 + \left(130 a^{2} + 34 a + 87\right)\cdot 151 + \left(103 a^{2} + 116 a + 119\right)\cdot 151^{2} + \left(34 a^{2} + 140 a + 123\right)\cdot 151^{3} + \left(22 a^{2} + 97 a + 14\right)\cdot 151^{4} + \left(36 a^{2} + 141 a + 24\right)\cdot 151^{5} + \left(116 a^{2} + 116 a + 27\right)\cdot 151^{6} + \left(87 a^{2} + 45 a + 8\right)\cdot 151^{7} + \left(26 a^{2} + 62 a + 68\right)\cdot 151^{8} + \left(19 a^{2} + 31 a + 113\right)\cdot 151^{9} + \left(54 a^{2} + 65 a + 136\right)\cdot 151^{10} +O\left(151^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 134 a^{2} + 137 a + 39 + \left(56 a^{2} + 89 a + 88\right)\cdot 151 + \left(58 a^{2} + 58 a + 139\right)\cdot 151^{2} + \left(2 a^{2} + 111 a + 51\right)\cdot 151^{3} + \left(140 a^{2} + 94 a + 93\right)\cdot 151^{4} + \left(55 a^{2} + 84 a + 87\right)\cdot 151^{5} + \left(68 a^{2} + 118 a + 45\right)\cdot 151^{6} + \left(25 a^{2} + 78 a + 67\right)\cdot 151^{7} + \left(16 a^{2} + 59 a + 111\right)\cdot 151^{8} + \left(a^{2} + 46 a\right)\cdot 151^{9} + \left(83 a^{2} + 13 a + 5\right)\cdot 151^{10} +O\left(151^{ 11 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 149 a^{2} + 104 a + 49 + \left(57 a^{2} + 30 a + 139\right)\cdot 151 + \left(90 a^{2} + 84 a + 9\right)\cdot 151^{2} + \left(32 a^{2} + 138 a + 72\right)\cdot 151^{3} + \left(28 a^{2} + 87 a + 119\right)\cdot 151^{4} + \left(15 a^{2} + 112 a + 110\right)\cdot 151^{5} + \left(25 a^{2} + 19 a + 16\right)\cdot 151^{6} + \left(67 a^{2} + 71 a + 95\right)\cdot 151^{7} + \left(6 a^{2} + 146 a + 54\right)\cdot 151^{8} + \left(122 a^{2} + 47 a + 81\right)\cdot 151^{9} + \left(91 a^{2} + 34 a + 111\right)\cdot 151^{10} +O\left(151^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,2,3)(4,9,7)$ |
| $(1,5,7)(2,6,9)(3,8,4)$ |
| $(2,3)(5,6)(7,9)$ |
| $(4,8)(5,7)(6,9)$ |
| $(4,7,9)(5,8,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(2,3)(4,9)(5,8)$ | $0$ |
| $9$ | $2$ | $(4,8)(5,7)(6,9)$ | $-2$ |
| $9$ | $2$ | $(1,8)(2,6)(3,5)(7,9)$ | $0$ |
| $2$ | $3$ | $(1,3,2)(4,9,7)(5,8,6)$ | $-3$ |
| $6$ | $3$ | $(1,2,3)(4,9,7)$ | $0$ |
| $6$ | $3$ | $(1,7,6)(2,9,8)(3,4,5)$ | $0$ |
| $12$ | $3$ | $(1,5,7)(2,6,9)(3,8,4)$ | $0$ |
| $18$ | $6$ | $(1,6,7)(2,5,9,3,8,4)$ | $0$ |
| $18$ | $6$ | $(1,2,3)(4,8,9,6,7,5)$ | $1$ |
| $18$ | $6$ | $(1,6,3,8,2,5)(7,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
