Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 16.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$: $ x^{3} + 3 x + 129 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 + 75\cdot 131 + 27\cdot 131^{2} + 25\cdot 131^{3} + 51\cdot 131^{4} + 33\cdot 131^{5} + 36\cdot 131^{6} + 44\cdot 131^{7} + 64\cdot 131^{8} + 53\cdot 131^{9} + 102\cdot 131^{10} + 124\cdot 131^{11} + 110\cdot 131^{12} + 129\cdot 131^{13} + 89\cdot 131^{14} + 102\cdot 131^{15} +O\left(131^{ 16 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 + 83\cdot 131 + 128\cdot 131^{2} + 23\cdot 131^{3} + 36\cdot 131^{4} + 35\cdot 131^{5} + 94\cdot 131^{6} + 64\cdot 131^{7} + 10\cdot 131^{8} + 3\cdot 131^{9} + 7\cdot 131^{10} + 107\cdot 131^{11} + 77\cdot 131^{12} + 115\cdot 131^{13} + 65\cdot 131^{14} + 68\cdot 131^{15} +O\left(131^{ 16 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 92 + 120\cdot 131 + 86\cdot 131^{2} + 73\cdot 131^{3} + 70\cdot 131^{4} + 42\cdot 131^{5} + 42\cdot 131^{6} + 94\cdot 131^{7} + 88\cdot 131^{8} + 53\cdot 131^{9} + 8\cdot 131^{10} + 61\cdot 131^{11} + 129\cdot 131^{12} + 87\cdot 131^{13} + 25\cdot 131^{14} + 115\cdot 131^{15} +O\left(131^{ 16 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a^{2} + 51 a + 11 + \left(13 a^{2} + 50 a + 44\right)\cdot 131 + \left(107 a^{2} + 16 a + 54\right)\cdot 131^{2} + \left(a^{2} + 18 a + 94\right)\cdot 131^{3} + \left(55 a^{2} + 44 a + 88\right)\cdot 131^{4} + \left(105 a^{2} + 30 a + 68\right)\cdot 131^{5} + \left(2 a^{2} + 27 a + 66\right)\cdot 131^{6} + \left(18 a^{2} + 37 a + 115\right)\cdot 131^{7} + \left(47 a^{2} + 110 a + 14\right)\cdot 131^{8} + \left(81 a^{2} + 94 a + 47\right)\cdot 131^{9} + \left(118 a^{2} + 16 a + 93\right)\cdot 131^{10} + \left(63 a^{2} + 87 a + 117\right)\cdot 131^{11} + \left(10 a^{2} + 60 a + 14\right)\cdot 131^{12} + \left(96 a^{2} + 121 a + 102\right)\cdot 131^{13} + \left(80 a^{2} + 29 a + 96\right)\cdot 131^{14} + \left(64 a^{2} + 54 a + 34\right)\cdot 131^{15} +O\left(131^{ 16 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 a^{2} + 64 a + 54 + \left(28 a^{2} + 53 a + 76\right)\cdot 131 + \left(13 a^{2} + 51 a + 61\right)\cdot 131^{2} + \left(5 a^{2} + 105 a + 9\right)\cdot 131^{3} + \left(123 a^{2} + 21 a + 40\right)\cdot 131^{4} + \left(97 a^{2} + 103 a + 126\right)\cdot 131^{5} + \left(76 a^{2} + 79 a + 34\right)\cdot 131^{6} + \left(34 a^{2} + 74 a + 9\right)\cdot 131^{7} + \left(16 a^{2} + 15 a + 101\right)\cdot 131^{8} + \left(55 a^{2} + 34 a + 101\right)\cdot 131^{9} + \left(7 a^{2} + 59 a + 75\right)\cdot 131^{10} + \left(122 a^{2} + 79 a + 25\right)\cdot 131^{11} + \left(13 a^{2} + 100 a + 15\right)\cdot 131^{12} + \left(116 a^{2} + 55 a + 80\right)\cdot 131^{13} + \left(2 a^{2} + 11 a + 53\right)\cdot 131^{14} + \left(73 a^{2} + 101 a + 101\right)\cdot 131^{15} +O\left(131^{ 16 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 42 a^{2} + 120 a + 80 + \left(52 a^{2} + 77 a + 124\right)\cdot 131 + \left(89 a^{2} + 117 a + 82\right)\cdot 131^{2} + \left(78 a^{2} + 125 a + 25\right)\cdot 131^{3} + \left(105 a^{2} + 104 a + 5\right)\cdot 131^{4} + \left(5 a^{2} + 120 a + 73\right)\cdot 131^{5} + \left(50 a^{2} + 95 a + 112\right)\cdot 131^{6} + \left(73 a^{2} + 86\right)\cdot 131^{7} + \left(123 a^{2} + 26 a + 53\right)\cdot 131^{8} + \left(10 a^{2} + 74 a + 13\right)\cdot 131^{9} + \left(125 a^{2} + 59 a + 49\right)\cdot 131^{10} + \left(32 a^{2} + 81 a + 109\right)\cdot 131^{11} + \left(44 a^{2} + 62 a + 75\right)\cdot 131^{12} + \left(124 a^{2} + 43 a + 96\right)\cdot 131^{13} + \left(38 a^{2} + 84 a + 125\right)\cdot 131^{14} + \left(103 a^{2} + 73 a + 30\right)\cdot 131^{15} +O\left(131^{ 16 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 55 a^{2} + 129 a + 113 + \left(76 a^{2} + 108 a + 39\right)\cdot 131 + \left(51 a^{2} + 85 a + 74\right)\cdot 131^{2} + \left(82 a^{2} + 111 a + 124\right)\cdot 131^{3} + \left(62 a^{2} + 31 a + 103\right)\cdot 131^{4} + \left(63 a^{2} + 118 a + 115\right)\cdot 131^{5} + \left(92 a^{2} + 84 a + 114\right)\cdot 131^{6} + \left(40 a^{2} + 20 a + 29\right)\cdot 131^{7} + \left(69 a^{2} + 127 a + 59\right)\cdot 131^{8} + \left(18 a^{2} + 28 a + 52\right)\cdot 131^{9} + \left(34 a^{2} + 31 a + 55\right)\cdot 131^{10} + \left(53 a^{2} + 121 a + 96\right)\cdot 131^{11} + \left(19 a^{2} + 10 a + 32\right)\cdot 131^{12} + \left(126 a^{2} + 48 a + 31\right)\cdot 131^{13} + \left(129 a^{2} + 36 a + 64\right)\cdot 131^{14} + \left(100 a^{2} + 14 a + 107\right)\cdot 131^{15} +O\left(131^{ 16 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 60 a^{2} + 78 a + 116 + \left(50 a^{2} + 130 a + 120\right)\cdot 131 + \left(28 a^{2} + 92 a + 91\right)\cdot 131^{2} + \left(47 a^{2} + 30 a + 93\right)\cdot 131^{3} + \left(33 a^{2} + 4 a + 122\right)\cdot 131^{4} + \left(27 a^{2} + 38 a + 115\right)\cdot 131^{5} + \left(4 a^{2} + 86 a + 20\right)\cdot 131^{6} + \left(23 a^{2} + 55 a + 117\right)\cdot 131^{7} + \left(122 a^{2} + 89 a + 50\right)\cdot 131^{8} + \left(64 a^{2} + 22 a + 121\right)\cdot 131^{9} + \left(129 a^{2} + 12 a + 57\right)\cdot 131^{10} + \left(106 a^{2} + 101 a + 126\right)\cdot 131^{11} + \left(72 a^{2} + 98 a + 1\right)\cdot 131^{12} + \left(21 a^{2} + 31 a + 22\right)\cdot 131^{13} + \left(89 a^{2} + 35 a + 95\right)\cdot 131^{14} + \left(85 a^{2} + 87 a + 126\right)\cdot 131^{15} +O\left(131^{ 16 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 72 a^{2} + 82 a + 16 + \left(41 a^{2} + 102 a + 101\right)\cdot 131 + \left(103 a^{2} + 28 a + 46\right)\cdot 131^{2} + \left(46 a^{2} + a + 53\right)\cdot 131^{3} + \left(13 a^{2} + 55 a + 5\right)\cdot 131^{4} + \left(93 a^{2} + 113 a + 44\right)\cdot 131^{5} + \left(35 a^{2} + 18 a + 1\right)\cdot 131^{6} + \left(72 a^{2} + 73 a + 93\right)\cdot 131^{7} + \left(14 a^{2} + 24 a + 80\right)\cdot 131^{8} + \left(31 a^{2} + 7 a + 77\right)\cdot 131^{9} + \left(109 a^{2} + 83 a + 74\right)\cdot 131^{10} + \left(13 a^{2} + 53 a + 17\right)\cdot 131^{11} + \left(101 a^{2} + 59 a + 65\right)\cdot 131^{12} + \left(39 a^{2} + 92 a + 120\right)\cdot 131^{13} + \left(51 a^{2} + 64 a + 37\right)\cdot 131^{14} + \left(96 a^{2} + 62 a + 98\right)\cdot 131^{15} +O\left(131^{ 16 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,8)(2,5)(3,6)$ |
| $(5,6,8)$ |
| $(4,9,7)$ |
| $(4,8,9,5,7,6)$ |
| $(1,3,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $9$ | $2$ | $(1,8)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,2,3)(4,9,7)(5,6,8)$ | $3 \zeta_{3}$ |
| $1$ | $3$ | $(1,3,2)(4,7,9)(5,8,6)$ | $-3 \zeta_{3} - 3$ |
| $3$ | $3$ | $(1,3,2)$ | $2 \zeta_{3} + 1$ |
| $3$ | $3$ | $(1,2,3)$ | $-2 \zeta_{3} - 1$ |
| $3$ | $3$ | $(1,2,3)(5,6,8)$ | $-\zeta_{3} - 2$ |
| $3$ | $3$ | $(1,3,2)(5,8,6)$ | $\zeta_{3} - 1$ |
| $3$ | $3$ | $(1,2,3)(4,7,9)(5,6,8)$ | $\zeta_{3} + 2$ |
| $3$ | $3$ | $(1,3,2)(4,9,7)(5,8,6)$ | $-\zeta_{3} + 1$ |
| $6$ | $3$ | $(1,3,2)(4,9,7)$ | $0$ |
| $18$ | $3$ | $(1,4,8)(2,9,5)(3,7,6)$ | $0$ |
| $9$ | $6$ | $(1,5,2,6,3,8)$ | $\zeta_{3} + 1$ |
| $9$ | $6$ | $(1,8,3,6,2,5)$ | $-\zeta_{3}$ |
| $9$ | $6$ | $(1,8)(2,5)(3,6)(4,9,7)$ | $-1$ |
| $9$ | $6$ | $(1,8)(2,5)(3,6)(4,7,9)$ | $-1$ |
| $9$ | $6$ | $(1,5,2,6,3,8)(4,9,7)$ | $\zeta_{3} + 1$ |
| $9$ | $6$ | $(1,8,3,6,2,5)(4,7,9)$ | $-\zeta_{3}$ |
| $9$ | $6$ | $(1,6,3,5,2,8)(4,9,7)$ | $-\zeta_{3}$ |
| $9$ | $6$ | $(1,8,2,5,3,6)(4,7,9)$ | $\zeta_{3} + 1$ |
| $18$ | $9$ | $(1,9,5,2,7,6,3,4,8)$ | $0$ |
| $18$ | $9$ | $(1,5,7,3,8,9,2,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
