Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 15.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$: $ x^{3} + 3 x + 129 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 34 + 16\cdot 131 + 94\cdot 131^{2} + 52\cdot 131^{3} + 85\cdot 131^{4} + 120\cdot 131^{5} + 71\cdot 131^{6} + 39\cdot 131^{7} + 54\cdot 131^{8} + 108\cdot 131^{9} + 49\cdot 131^{10} + 71\cdot 131^{11} + 107\cdot 131^{12} + 88\cdot 131^{13} + 102\cdot 131^{14} +O\left(131^{ 15 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 51 + 127\cdot 131 + 40\cdot 131^{2} + 67\cdot 131^{3} + 36\cdot 131^{4} + 109\cdot 131^{5} + 13\cdot 131^{6} + 122\cdot 131^{7} + 43\cdot 131^{8} + 77\cdot 131^{9} + 100\cdot 131^{10} + 2\cdot 131^{11} + 119\cdot 131^{12} + 60\cdot 131^{13} + 84\cdot 131^{14} +O\left(131^{ 15 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 72 + 11\cdot 131 + 123\cdot 131^{2} + 36\cdot 131^{3} + 31\cdot 131^{4} + 127\cdot 131^{5} + 56\cdot 131^{6} + 17\cdot 131^{7} + 122\cdot 131^{8} + 81\cdot 131^{9} + 125\cdot 131^{10} + 49\cdot 131^{11} + 100\cdot 131^{12} + 101\cdot 131^{13} + 54\cdot 131^{14} +O\left(131^{ 15 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a^{2} + 4 a + 99 + \left(97 a^{2} + 15 a + 58\right)\cdot 131 + \left(21 a^{2} + 99 a + 35\right)\cdot 131^{2} + \left(3 a^{2} + 130 a + 121\right)\cdot 131^{3} + \left(30 a^{2} + 126 a + 6\right)\cdot 131^{4} + \left(14 a^{2} + 85 a + 44\right)\cdot 131^{5} + \left(46 a^{2} + 61 a + 97\right)\cdot 131^{6} + \left(51 a^{2} + 47 a + 114\right)\cdot 131^{7} + \left(48 a^{2} + 89 a + 91\right)\cdot 131^{8} + \left(77 a^{2} + 107 a + 17\right)\cdot 131^{9} + \left(16 a^{2} + 74 a + 49\right)\cdot 131^{10} + \left(110 a^{2} + 83 a + 41\right)\cdot 131^{11} + \left(42 a^{2} + 75 a + 92\right)\cdot 131^{12} + \left(18 a^{2} + 86 a + 25\right)\cdot 131^{13} + \left(7 a^{2} + 72 a + 40\right)\cdot 131^{14} +O\left(131^{ 15 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 50 a^{2} + 42 a + 54 + \left(25 a^{2} + 36 a + 46\right)\cdot 131 + \left(71 a^{2} + 8 a + 3\right)\cdot 131^{2} + \left(119 a^{2} + 9 a + 92\right)\cdot 131^{3} + \left(27 a^{2} + 21 a + 2\right)\cdot 131^{4} + \left(45 a^{2} + 71 a + 106\right)\cdot 131^{5} + \left(48 a^{2} + 10 a + 101\right)\cdot 131^{6} + \left(62 a^{2} + 108 a + 5\right)\cdot 131^{7} + \left(113 a^{2} + 124 a + 91\right)\cdot 131^{8} + \left(115 a^{2} + 103 a + 94\right)\cdot 131^{9} + \left(53 a^{2} + 4 a + 123\right)\cdot 131^{10} + \left(8 a^{2} + 42 a + 99\right)\cdot 131^{11} + \left(49 a^{2} + 106 a + 104\right)\cdot 131^{12} + \left(31 a^{2} + 64 a + 51\right)\cdot 131^{13} + \left(42 a^{2} + 117 a + 110\right)\cdot 131^{14} +O\left(131^{ 15 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 55 a^{2} + 108 a + 104 + \left(103 a^{2} + 95 a + 115\right)\cdot 131 + \left(13 a^{2} + 19 a + 36\right)\cdot 131^{2} + \left(8 a^{2} + 35 a + 67\right)\cdot 131^{3} + \left(32 a^{2} + 55 a + 22\right)\cdot 131^{4} + \left(111 a^{2} + 24 a + 44\right)\cdot 131^{5} + \left(15 a^{2} + 64 a + 66\right)\cdot 131^{6} + \left(116 a^{2} + 22 a + 29\right)\cdot 131^{7} + \left(85 a^{2} + 94 a + 103\right)\cdot 131^{8} + \left(81 a^{2} + 106 a + 123\right)\cdot 131^{9} + \left(26 a^{2} + 106 a + 119\right)\cdot 131^{10} + \left(46 a^{2} + 84 a + 98\right)\cdot 131^{11} + \left(37 a^{2} + 43 a + 2\right)\cdot 131^{12} + \left(47 a^{2} + 120 a + 109\right)\cdot 131^{13} + \left(98 a^{2} + 30 a + 2\right)\cdot 131^{14} +O\left(131^{ 15 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 74 a^{2} + 85 a + 102 + \left(8 a^{2} + 79 a + 12\right)\cdot 131 + \left(38 a^{2} + 23 a + 68\right)\cdot 131^{2} + \left(8 a^{2} + 122 a\right)\cdot 131^{3} + \left(73 a^{2} + 113 a + 93\right)\cdot 131^{4} + \left(71 a^{2} + 104 a + 27\right)\cdot 131^{5} + \left(36 a^{2} + 58 a + 78\right)\cdot 131^{6} + \left(17 a^{2} + 106 a + 46\right)\cdot 131^{7} + \left(100 a^{2} + 47 a + 64\right)\cdot 131^{8} + \left(68 a^{2} + 50 a\right)\cdot 131^{9} + \left(60 a^{2} + 51 a + 6\right)\cdot 131^{10} + \left(12 a^{2} + 5 a + 108\right)\cdot 131^{11} + \left(39 a^{2} + 80 a + 84\right)\cdot 131^{12} + \left(81 a^{2} + 110 a + 20\right)\cdot 131^{13} + \left(81 a^{2} + 71 a + 58\right)\cdot 131^{14} +O\left(131^{ 15 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 88 a^{2} + 80 a + 39 + \left(129 a^{2} + 125 a + 37\right)\cdot 131 + \left(37 a^{2} + 52 a + 85\right)\cdot 131^{2} + \left(10 a^{2} + 66 a + 71\right)\cdot 131^{3} + \left(3 a^{2} + 74 a + 95\right)\cdot 131^{4} + \left(44 a^{2} + 75 a + 40\right)\cdot 131^{5} + \left(66 a^{2} + 40 a + 36\right)\cdot 131^{6} + \left(92 a^{2} + 22 a + 113\right)\cdot 131^{7} + \left(41 a^{2} + 113 a + 14\right)\cdot 131^{8} + \left(54 a^{2} + 92 a + 69\right)\cdot 131^{9} + \left(24 a^{2} + 2 a + 115\right)\cdot 131^{10} + \left(106 a^{2} + 3 a + 87\right)\cdot 131^{11} + \left(40 a^{2} + 88 a + 9\right)\cdot 131^{12} + \left(21 a^{2} + 65 a + 57\right)\cdot 131^{13} + \left(34 a^{2} + 70 a + 5\right)\cdot 131^{14} +O\left(131^{ 15 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 119 a^{2} + 74 a + 101 + \left(28 a^{2} + 40 a + 97\right)\cdot 131 + \left(79 a^{2} + 58 a + 36\right)\cdot 131^{2} + \left(112 a^{2} + 29 a + 14\right)\cdot 131^{3} + \left(95 a^{2} + a + 19\right)\cdot 131^{4} + \left(106 a^{2} + 31 a + 35\right)\cdot 131^{5} + \left(48 a^{2} + 26 a + 1\right)\cdot 131^{6} + \left(53 a^{2} + 86 a + 35\right)\cdot 131^{7} + \left(3 a^{2} + 54 a + 69\right)\cdot 131^{8} + \left(126 a^{2} + 62 a + 81\right)\cdot 131^{9} + \left(79 a^{2} + 21 a + 95\right)\cdot 131^{10} + \left(109 a^{2} + 43 a + 94\right)\cdot 131^{11} + \left(52 a^{2} + 130 a + 33\right)\cdot 131^{12} + \left(62 a^{2} + 75 a + 8\right)\cdot 131^{13} + \left(129 a^{2} + 29 a + 65\right)\cdot 131^{14} +O\left(131^{ 15 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(6,9,8)$ |
| $(4,5,7)$ |
| $(1,3,2)$ |
| $(1,7)(2,4)(3,5)(6,9,8)$ |
| $(1,2,3)(4,6,7,9,5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $9$ | $2$ | $(1,7)(2,4)(3,5)$ | $1$ |
| $1$ | $3$ | $(1,3,2)(4,7,5)(6,9,8)$ | $-3 \zeta_{3} - 3$ |
| $1$ | $3$ | $(1,2,3)(4,5,7)(6,8,9)$ | $3 \zeta_{3}$ |
| $3$ | $3$ | $(1,3,2)$ | $-\zeta_{3} - 2$ |
| $3$ | $3$ | $(1,2,3)$ | $\zeta_{3} - 1$ |
| $3$ | $3$ | $(1,3,2)(4,7,5)(6,8,9)$ | $2 \zeta_{3} + 1$ |
| $3$ | $3$ | $(1,2,3)(4,5,7)(6,9,8)$ | $-2 \zeta_{3} - 1$ |
| $3$ | $3$ | $(1,3,2)(4,7,5)$ | $\zeta_{3} + 2$ |
| $3$ | $3$ | $(1,2,3)(4,5,7)$ | $-\zeta_{3} + 1$ |
| $6$ | $3$ | $(1,3,2)(4,5,7)$ | $0$ |
| $18$ | $3$ | $(1,9,4)(2,6,5)(3,8,7)$ | $0$ |
| $9$ | $6$ | $(1,7)(2,4)(3,5)(6,9,8)$ | $\zeta_{3}$ |
| $9$ | $6$ | $(1,7)(2,4)(3,5)(6,8,9)$ | $-\zeta_{3} - 1$ |
| $9$ | $6$ | $(1,2,3)(4,6,7,9,5,8)$ | $\zeta_{3}$ |
| $9$ | $6$ | $(1,3,2)(4,8,5,9,7,6)$ | $-\zeta_{3} - 1$ |
| $9$ | $6$ | $(1,5,3,4,2,7)(6,9,8)$ | $1$ |
| $9$ | $6$ | $(1,7,2,4,3,5)(6,8,9)$ | $1$ |
| $9$ | $6$ | $(4,8,5,9,7,6)$ | $\zeta_{3}$ |
| $9$ | $6$ | $(4,6,7,9,5,8)$ | $-\zeta_{3} - 1$ |
| $18$ | $9$ | $(1,9,4,3,8,7,2,6,5)$ | $0$ |
| $18$ | $9$ | $(1,4,8,2,5,9,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
