Basic invariants
Galois action
Roots of defining polynomial
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 + 78\cdot 151 + 82\cdot 151^{2} + 132\cdot 151^{3} + 15\cdot 151^{4} + 79\cdot 151^{6} + 142\cdot 151^{7} + 126\cdot 151^{8} + 97\cdot 151^{9} +O\left(151^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 40 + 151 + 143\cdot 151^{2} + 105\cdot 151^{3} + 107\cdot 151^{4} + 138\cdot 151^{5} + 57\cdot 151^{6} + 94\cdot 151^{7} + 73\cdot 151^{8} + 100\cdot 151^{9} +O\left(151^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 69 + 39\cdot 151 + 95\cdot 151^{2} + 83\cdot 151^{3} + 109\cdot 151^{4} + 10\cdot 151^{5} + 42\cdot 151^{6} + 37\cdot 151^{7} + 127\cdot 151^{8} + 64\cdot 151^{9} +O\left(151^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 65 a^{2} + 4 a + 141 + \left(84 a^{2} + 38 a + 46\right)\cdot 151 + \left(141 a^{2} + 2 a + 66\right)\cdot 151^{2} + \left(50 a^{2} + 78 a + 88\right)\cdot 151^{3} + \left(46 a^{2} + 60 a + 87\right)\cdot 151^{4} + \left(135 a^{2} + 134 a + 43\right)\cdot 151^{5} + \left(12 a^{2} + 46 a + 51\right)\cdot 151^{6} + \left(79 a^{2} + 29\right)\cdot 151^{7} + \left(20 a^{2} + 37 a + 118\right)\cdot 151^{8} + \left(99 a^{2} + 43 a + 115\right)\cdot 151^{9} +O\left(151^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 90 a^{2} + 61 a + 125 + \left(34 a^{2} + 125 a + 143\right)\cdot 151 + \left(148 a^{2} + 44 a + 19\right)\cdot 151^{2} + \left(82 a^{2} + 17 a + 145\right)\cdot 151^{3} + \left(129 a^{2} + 126 a + 102\right)\cdot 151^{4} + \left(108 a^{2} + 65 a + 119\right)\cdot 151^{5} + \left(141 a^{2} + 135 a + 92\right)\cdot 151^{6} + \left(20 a^{2} + 144 a + 46\right)\cdot 151^{7} + \left(77 a^{2} + 108 a + 89\right)\cdot 151^{8} + \left(125 a^{2} + 141 a + 147\right)\cdot 151^{9} +O\left(151^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 101 a^{2} + 34 a + 14 + \left(68 a^{2} + 15 a + 137\right)\cdot 151 + \left(11 a^{2} + 43 a + 29\right)\cdot 151^{2} + \left(39 a^{2} + 89 a + 30\right)\cdot 151^{3} + \left(85 a^{2} + 64 a + 63\right)\cdot 151^{4} + \left(119 a^{2} + 25 a + 83\right)\cdot 151^{5} + \left(111 a^{2} + 70 a + 16\right)\cdot 151^{6} + \left(36 a^{2} + 137 a + 1\right)\cdot 151^{7} + \left(149 a^{2} + 13 a + 53\right)\cdot 151^{8} + \left(58 a^{2} + 89 a + 139\right)\cdot 151^{9} +O\left(151^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 104 a^{2} + 3 a + 84 + \left(139 a^{2} + 53 a + 12\right)\cdot 151 + \left(125 a^{2} + 88 a + 5\right)\cdot 151^{2} + \left(77 a^{2} + 25 a + 41\right)\cdot 151^{3} + \left(11 a^{2} + 90 a + 24\right)\cdot 151^{4} + \left(91 a^{2} + 19 a + 7\right)\cdot 151^{5} + \left(35 a^{2} + 76 a + 22\right)\cdot 151^{6} + \left(7 a^{2} + 25 a + 138\right)\cdot 151^{7} + \left(29 a^{2} + 94 a + 6\right)\cdot 151^{8} + \left(75 a^{2} + 134 a + 114\right)\cdot 151^{9} +O\left(151^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 108 a^{2} + 87 a + 137 + \left(127 a^{2} + 123 a + 54\right)\cdot 151 + \left(27 a^{2} + 17 a + 40\right)\cdot 151^{2} + \left(141 a^{2} + 108 a + 83\right)\cdot 151^{3} + \left(9 a^{2} + 85 a + 73\right)\cdot 151^{4} + \left(102 a^{2} + 65 a + 14\right)\cdot 151^{5} + \left(124 a^{2} + 90 a + 31\right)\cdot 151^{6} + \left(122 a^{2} + 131 a + 64\right)\cdot 151^{7} + \left(44 a^{2} + 98 a + 17\right)\cdot 151^{8} + \left(101 a^{2} + 25 a + 81\right)\cdot 151^{9} +O\left(151^{ 10 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 136 a^{2} + 113 a + 138 + \left(148 a^{2} + 97 a + 89\right)\cdot 151 + \left(148 a^{2} + 105 a + 121\right)\cdot 151^{2} + \left(60 a^{2} + 134 a + 44\right)\cdot 151^{3} + \left(19 a^{2} + 25 a + 19\right)\cdot 151^{4} + \left(47 a^{2} + 142 a + 35\right)\cdot 151^{5} + \left(26 a^{2} + 33 a + 60\right)\cdot 151^{6} + \left(35 a^{2} + 13 a + 50\right)\cdot 151^{7} + \left(132 a^{2} + 100 a + 142\right)\cdot 151^{8} + \left(143 a^{2} + 18 a + 44\right)\cdot 151^{9} +O\left(151^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(4,6,9)(5,8,7)$ |
| $(1,9,8)(2,4,5)(3,6,7)$ |
| $(1,2,3)(4,9,6)$ |
| $(2,3)(4,6)(5,7)$ |
| $(1,9)(2,4)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(2,3)(4,6)(5,7)$ |
$0$ |
| $9$ |
$2$ |
$(1,9)(2,4)(3,6)$ |
$-2$ |
| $9$ |
$2$ |
$(1,9)(2,6)(3,4)(5,7)$ |
$0$ |
| $2$ |
$3$ |
$(1,2,3)(4,6,9)(5,7,8)$ |
$-3$ |
| $6$ |
$3$ |
$(1,9,8)(2,4,5)(3,6,7)$ |
$0$ |
| $6$ |
$3$ |
$(4,6,9)(5,8,7)$ |
$0$ |
| $12$ |
$3$ |
$(1,6,7)(2,9,8)(3,4,5)$ |
$0$ |
| $18$ |
$6$ |
$(1,9,8)(2,6,5,3,4,7)$ |
$0$ |
| $18$ |
$6$ |
$(1,4,2,6,3,9)(5,8,7)$ |
$1$ |
| $18$ |
$6$ |
$(1,4,3,6,2,9)(7,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
