Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 13.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$: $ x^{3} + 3 x + 129 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 33 + 108\cdot 131 + 123\cdot 131^{2} + 102\cdot 131^{3} + 74\cdot 131^{4} + 35\cdot 131^{5} + 43\cdot 131^{6} + 91\cdot 131^{7} + 80\cdot 131^{8} + 38\cdot 131^{9} + 29\cdot 131^{10} + 16\cdot 131^{11} + 46\cdot 131^{12} +O\left(131^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 50 + 117\cdot 131 + 101\cdot 131^{2} + 38\cdot 131^{3} + 57\cdot 131^{4} + 65\cdot 131^{5} + 36\cdot 131^{6} + 128\cdot 131^{7} + 37\cdot 131^{8} + 96\cdot 131^{9} + 69\cdot 131^{10} + 119\cdot 131^{11} + 2\cdot 131^{12} +O\left(131^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 103 + 36\cdot 131 + 119\cdot 131^{2} + 100\cdot 131^{3} + 78\cdot 131^{4} + 104\cdot 131^{5} + 43\cdot 131^{6} + 103\cdot 131^{7} + 128\cdot 131^{8} + 73\cdot 131^{9} + 27\cdot 131^{10} + 6\cdot 131^{11} + 58\cdot 131^{12} +O\left(131^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 a^{2} + 89 a + 48 + \left(107 a^{2} + 54 a + 29\right)\cdot 131 + \left(84 a^{2} + 46 a + 126\right)\cdot 131^{2} + \left(41 a^{2} + 117 a + 4\right)\cdot 131^{3} + \left(88 a^{2} + 23 a + 55\right)\cdot 131^{4} + \left(5 a^{2} + 61 a + 27\right)\cdot 131^{5} + \left(32 a^{2} + 50 a + 67\right)\cdot 131^{6} + \left(54 a^{2} + 85 a + 69\right)\cdot 131^{7} + \left(57 a^{2} + 58 a + 114\right)\cdot 131^{8} + \left(70 a^{2} + 119 a + 49\right)\cdot 131^{9} + \left(a^{2} + 52 a + 70\right)\cdot 131^{10} + \left(114 a^{2} + 130 a + 47\right)\cdot 131^{11} + \left(12 a^{2} + 47 a + 18\right)\cdot 131^{12} +O\left(131^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 a^{2} + 118 a + 21 + \left(81 a + 54\right)\cdot 131 + \left(21 a^{2} + 121 a + 101\right)\cdot 131^{2} + \left(127 a^{2} + 54 a + 120\right)\cdot 131^{3} + \left(5 a^{2} + 52 a + 106\right)\cdot 131^{4} + \left(72 a^{2} + 122 a + 15\right)\cdot 131^{5} + \left(57 a^{2} + 108 a + 27\right)\cdot 131^{6} + \left(121 a^{2} + 72 a + 43\right)\cdot 131^{7} + \left(96 a^{2} + 66 a + 24\right)\cdot 131^{8} + \left(68 a^{2} + 49 a + 115\right)\cdot 131^{9} + \left(118 a^{2} + 113 a + 83\right)\cdot 131^{10} + \left(54 a^{2} + 4 a + 24\right)\cdot 131^{11} + \left(37 a^{2} + 61 a + 90\right)\cdot 131^{12} +O\left(131^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 49 a^{2} + 24 a + 126 + \left(41 a^{2} + 17 a + 28\right)\cdot 131 + \left(46 a^{2} + 49 a + 49\right)\cdot 131^{2} + \left(99 a^{2} + 106 a + 120\right)\cdot 131^{3} + \left(80 a^{2} + 91 a + 39\right)\cdot 131^{4} + \left(110 a^{2} + 49 a + 106\right)\cdot 131^{5} + \left(36 a^{2} + 25 a + 76\right)\cdot 131^{6} + \left(34 a^{2} + 51 a + 29\right)\cdot 131^{7} + \left(48 a^{2} + 21 a + 96\right)\cdot 131^{8} + \left(20 a^{2} + 35 a + 80\right)\cdot 131^{9} + \left(105 a^{2} + 53 a + 15\right)\cdot 131^{10} + \left(54 a^{2} + 43 a + 60\right)\cdot 131^{11} + \left(84 a^{2} + 25 a + 30\right)\cdot 131^{12} +O\left(131^{ 13 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 72 a^{2} + 18 a + 41 + \left(113 a^{2} + 59 a + 42\right)\cdot 131 + \left(130 a^{2} + 35 a + 87\right)\cdot 131^{2} + \left(120 a^{2} + 38 a + 32\right)\cdot 131^{3} + \left(92 a^{2} + 15 a + 64\right)\cdot 131^{4} + \left(14 a^{2} + 20 a + 45\right)\cdot 131^{5} + \left(62 a^{2} + 55 a + 127\right)\cdot 131^{6} + \left(42 a^{2} + 125 a + 45\right)\cdot 131^{7} + \left(25 a^{2} + 50 a + 50\right)\cdot 131^{8} + \left(40 a^{2} + 107 a + 120\right)\cdot 131^{9} + \left(24 a^{2} + 24 a + 115\right)\cdot 131^{10} + \left(93 a^{2} + 88 a + 5\right)\cdot 131^{11} + \left(33 a^{2} + 57 a + 60\right)\cdot 131^{12} +O\left(131^{ 13 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 102 a^{2} + 123 a + 28 + \left(98 a^{2} + 116 a + 120\right)\cdot 131 + \left(117 a + 60\right)\cdot 131^{2} + \left(11 a^{2} + 6 a + 19\right)\cdot 131^{3} + \left(71 a^{2} + 74 a + 106\right)\cdot 131^{4} + \left(88 a^{2} + 94 a + 48\right)\cdot 131^{5} + \left(36 a^{2} + 14 a + 116\right)\cdot 131^{6} + \left(27 a^{2} + 4 a + 116\right)\cdot 131^{7} + \left(49 a^{2} + 38 a + 59\right)\cdot 131^{8} + \left(78 a^{2} + 39 a + 3\right)\cdot 131^{9} + \left(122 a^{2} + 29 a + 92\right)\cdot 131^{10} + \left(28 a^{2} + 42 a + 103\right)\cdot 131^{11} + \left(82 a^{2} + 44 a + 48\right)\cdot 131^{12} +O\left(131^{ 13 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 127 a^{2} + 21 a + 78 + \left(31 a^{2} + 63 a + 117\right)\cdot 131 + \left(109 a^{2} + 22 a + 15\right)\cdot 131^{2} + \left(123 a^{2} + 69 a + 114\right)\cdot 131^{3} + \left(53 a^{2} + 4 a + 71\right)\cdot 131^{4} + \left(101 a^{2} + 45 a + 74\right)\cdot 131^{5} + \left(36 a^{2} + 7 a + 116\right)\cdot 131^{6} + \left(113 a^{2} + 54 a + 26\right)\cdot 131^{7} + \left(115 a^{2} + 26 a + 62\right)\cdot 131^{8} + \left(114 a^{2} + 42 a + 76\right)\cdot 131^{9} + \left(20 a^{2} + 119 a + 19\right)\cdot 131^{10} + \left(47 a^{2} + 83 a + 9\right)\cdot 131^{11} + \left(11 a^{2} + 25 a + 38\right)\cdot 131^{12} +O\left(131^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,5,6)(2,9,4)(3,8,7)$ |
| $(1,3,2)(4,7,6)$ |
| $(2,3)(4,5)(6,9)(7,8)$ |
| $(4,6,7)(5,9,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,7)(2,4)(3,6)(5,9)$ | $0$ |
| $2$ | $3$ | $(1,2,3)(4,7,6)(5,9,8)$ | $-3$ |
| $3$ | $3$ | $(1,3,2)(4,7,6)$ | $0$ |
| $3$ | $3$ | $(1,2,3)(4,6,7)$ | $0$ |
| $6$ | $3$ | $(1,5,6)(2,9,4)(3,8,7)$ | $0$ |
| $6$ | $3$ | $(1,6,5)(2,4,9)(3,7,8)$ | $0$ |
| $6$ | $3$ | $(1,9,6)(2,8,4)(3,5,7)$ | $0$ |
| $9$ | $6$ | $(1,4,3,7,2,6)(5,9)$ | $0$ |
| $9$ | $6$ | $(1,6,2,7,3,4)(5,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
