Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 139 }$ to precision 16.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 139 }$: $ x^{3} + 6 x + 137 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 36 + 101\cdot 139 + 11\cdot 139^{2} + 37\cdot 139^{3} + 62\cdot 139^{4} + 67\cdot 139^{5} + 31\cdot 139^{6} + 59\cdot 139^{7} + 125\cdot 139^{8} + 14\cdot 139^{9} + 2\cdot 139^{10} + 74\cdot 139^{11} + 65\cdot 139^{12} + 46\cdot 139^{13} + 95\cdot 139^{14} + 18\cdot 139^{15} +O\left(139^{ 16 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 68 + 94\cdot 139 + 88\cdot 139^{2} + 76\cdot 139^{3} + 79\cdot 139^{4} + 22\cdot 139^{5} + 67\cdot 139^{6} + 89\cdot 139^{7} + 18\cdot 139^{8} + 108\cdot 139^{10} + 28\cdot 139^{11} + 7\cdot 139^{12} + 40\cdot 139^{13} + 67\cdot 139^{14} + 4\cdot 139^{15} +O\left(139^{ 16 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 76 + 118\cdot 139 + 106\cdot 139^{2} + 112\cdot 139^{3} + 53\cdot 139^{4} + 19\cdot 139^{5} + 138\cdot 139^{6} + 23\cdot 139^{7} + 115\cdot 139^{8} + 93\cdot 139^{9} + 21\cdot 139^{10} + 133\cdot 139^{11} + 5\cdot 139^{12} + 62\cdot 139^{13} + 6\cdot 139^{14} + 3\cdot 139^{15} +O\left(139^{ 16 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a^{2} + 86 a + 88 + \left(64 a^{2} + 41 a + 66\right)\cdot 139 + \left(120 a^{2} + 61 a + 105\right)\cdot 139^{2} + \left(100 a^{2} + 82 a + 129\right)\cdot 139^{3} + \left(4 a^{2} + 137 a + 13\right)\cdot 139^{4} + \left(129 a^{2} + 49 a + 118\right)\cdot 139^{5} + \left(76 a^{2} + 61 a + 39\right)\cdot 139^{6} + \left(103 a^{2} + 91 a + 66\right)\cdot 139^{7} + \left(103 a^{2} + 133 a + 120\right)\cdot 139^{8} + \left(15 a^{2} + 137 a + 131\right)\cdot 139^{9} + \left(a^{2} + 4 a + 67\right)\cdot 139^{10} + \left(4 a^{2} + 78 a + 128\right)\cdot 139^{11} + \left(51 a^{2} + 86 a + 50\right)\cdot 139^{12} + \left(107 a^{2} + 48 a + 10\right)\cdot 139^{13} + \left(38 a^{2} + 121 a + 60\right)\cdot 139^{14} + \left(74 a^{2} + 85 a + 70\right)\cdot 139^{15} +O\left(139^{ 16 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 23 a^{2} + 110 a + 88 + \left(99 a^{2} + 26 a + 110\right)\cdot 139 + \left(31 a^{2} + 23 a + 109\right)\cdot 139^{2} + \left(80 a^{2} + 63 a + 55\right)\cdot 139^{3} + \left(69 a^{2} + 30 a + 125\right)\cdot 139^{4} + \left(121 a^{2} + 41 a + 59\right)\cdot 139^{5} + \left(126 a^{2} + 77 a + 1\right)\cdot 139^{6} + \left(124 a^{2} + 68 a + 95\right)\cdot 139^{7} + \left(123 a^{2} + 19 a + 8\right)\cdot 139^{8} + \left(72 a^{2} + 10 a + 1\right)\cdot 139^{9} + \left(116 a^{2} + 3 a + 127\right)\cdot 139^{10} + \left(120 a^{2} + 99 a + 106\right)\cdot 139^{11} + \left(121 a^{2} + 9 a + 104\right)\cdot 139^{12} + \left(91 a^{2} + 25 a + 134\right)\cdot 139^{13} + \left(17 a^{2} + 62 a + 62\right)\cdot 139^{14} + \left(93 a^{2} + 135 a + 34\right)\cdot 139^{15} +O\left(139^{ 16 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 24 a^{2} + 95 a + 92 + \left(40 a^{2} + 36 a + 13\right)\cdot 139 + \left(67 a^{2} + 128 a + 113\right)\cdot 139^{2} + \left(34 a^{2} + 92 a + 11\right)\cdot 139^{3} + \left(68 a^{2} + 35 a + 120\right)\cdot 139^{4} + \left(12 a^{2} + 132 a + 40\right)\cdot 139^{5} + \left(107 a^{2} + 64 a + 61\right)\cdot 139^{6} + \left(19 a^{2} + 55 a + 91\right)\cdot 139^{7} + \left(117 a^{2} + 80 a + 120\right)\cdot 139^{8} + \left(53 a^{2} + 12 a + 63\right)\cdot 139^{9} + \left(92 a^{2} + 2 a + 30\right)\cdot 139^{10} + \left(94 a^{2} + 63 a + 2\right)\cdot 139^{11} + \left(88 a^{2} + 71 a + 111\right)\cdot 139^{12} + \left(65 a^{2} + 79 a + 29\right)\cdot 139^{13} + \left(51 a^{2} + 79 a + 59\right)\cdot 139^{14} + \left(23 a^{2} + 51 a + 33\right)\cdot 139^{15} +O\left(139^{ 16 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 59 a^{2} + 120 a + 42 + \left(17 a^{2} + 19 a + 19\right)\cdot 139 + \left(96 a^{2} + 129 a + 8\right)\cdot 139^{2} + \left(3 a^{2} + 31 a + 19\right)\cdot 139^{3} + \left(13 a^{2} + 135 a + 47\right)\cdot 139^{4} + \left(12 a^{2} + 65 a + 67\right)\cdot 139^{5} + \left(127 a^{2} + 121 a + 101\right)\cdot 139^{6} + \left(32 a^{2} + a + 61\right)\cdot 139^{7} + \left(59 a^{2} + 82 a + 81\right)\cdot 139^{8} + \left(122 a^{2} + 69 a + 2\right)\cdot 139^{9} + \left(7 a^{2} + 136 a + 95\right)\cdot 139^{10} + \left(26 a^{2} + 52 a + 77\right)\cdot 139^{11} + \left(74 a^{2} + 125 a + 4\right)\cdot 139^{12} + \left(14 a^{2} + 10 a + 56\right)\cdot 139^{13} + \left(85 a^{2} + 122 a + 106\right)\cdot 139^{14} + \left(32 a^{2} + 32 a + 42\right)\cdot 139^{15} +O\left(139^{ 16 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 79 a^{2} + 72 a + 122 + \left(57 a^{2} + 77 a + 40\right)\cdot 139 + \left(61 a^{2} + 87 a + 8\right)\cdot 139^{2} + \left(34 a^{2} + 24 a + 3\right)\cdot 139^{3} + \left(121 a^{2} + 5 a + 63\right)\cdot 139^{4} + \left(136 a^{2} + 23 a + 10\right)\cdot 139^{5} + \left(73 a^{2} + 95 a + 28\right)\cdot 139^{6} + \left(2 a^{2} + 45 a + 79\right)\cdot 139^{7} + \left(115 a^{2} + 62 a + 26\right)\cdot 139^{8} + \left(70 a + 72\right)\cdot 139^{9} + \left(130 a^{2} + 136 a + 27\right)\cdot 139^{10} + \left(108 a^{2} + 7 a + 131\right)\cdot 139^{11} + \left(13 a^{2} + 66 a + 40\right)\cdot 139^{12} + \left(17 a^{2} + 79 a + 66\right)\cdot 139^{13} + \left(15 a^{2} + 34 a + 104\right)\cdot 139^{14} + \left(32 a^{2} + 20 a + 40\right)\cdot 139^{15} +O\left(139^{ 16 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 92 a^{2} + 73 a + 86 + \left(138 a^{2} + 75 a + 129\right)\cdot 139 + \left(39 a^{2} + 126 a + 3\right)\cdot 139^{2} + \left(24 a^{2} + 121 a + 110\right)\cdot 139^{3} + \left(a^{2} + 72 a + 129\right)\cdot 139^{4} + \left(5 a^{2} + 104 a + 10\right)\cdot 139^{5} + \left(44 a^{2} + 135 a + 87\right)\cdot 139^{6} + \left(133 a^{2} + 14 a + 128\right)\cdot 139^{7} + \left(36 a^{2} + 39 a + 77\right)\cdot 139^{8} + \left(12 a^{2} + 116 a + 36\right)\cdot 139^{9} + \left(69 a^{2} + 133 a + 76\right)\cdot 139^{10} + \left(62 a^{2} + 115 a + 12\right)\cdot 139^{11} + \left(67 a^{2} + 57 a + 26\right)\cdot 139^{12} + \left(120 a^{2} + 34 a + 110\right)\cdot 139^{13} + \left(69 a^{2} + 136 a + 132\right)\cdot 139^{14} + \left(22 a^{2} + 90 a + 29\right)\cdot 139^{15} +O\left(139^{ 16 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,2,3)(4,7,8)$ |
| $(1,5,3,6,2,9)(4,7,8)$ |
| $(4,8,7)(5,6,9)$ |
| $(4,8,7)$ |
| $(1,2,3)(4,6,8,5,7,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $9$ | $2$ | $(1,6)(2,5)(3,9)$ | $-1$ |
| $1$ | $3$ | $(1,3,2)(4,7,8)(5,6,9)$ | $3 \zeta_{3}$ |
| $1$ | $3$ | $(1,2,3)(4,8,7)(5,9,6)$ | $-3 \zeta_{3} - 3$ |
| $3$ | $3$ | $(1,3,2)(4,8,7)(5,6,9)$ | $\zeta_{3} + 2$ |
| $3$ | $3$ | $(1,2,3)(4,7,8)(5,9,6)$ | $-\zeta_{3} + 1$ |
| $3$ | $3$ | $(1,3,2)(5,6,9)$ | $-\zeta_{3} - 2$ |
| $3$ | $3$ | $(1,2,3)(5,9,6)$ | $\zeta_{3} - 1$ |
| $3$ | $3$ | $(4,8,7)$ | $2 \zeta_{3} + 1$ |
| $3$ | $3$ | $(4,7,8)$ | $-2 \zeta_{3} - 1$ |
| $6$ | $3$ | $(1,2,3)(4,7,8)$ | $0$ |
| $18$ | $3$ | $(1,7,5)(2,4,9)(3,8,6)$ | $0$ |
| $9$ | $6$ | $(1,5,3,6,2,9)(4,7,8)$ | $\zeta_{3} + 1$ |
| $9$ | $6$ | $(1,9,2,6,3,5)(4,8,7)$ | $-\zeta_{3}$ |
| $9$ | $6$ | $(1,6,3,9,2,5)$ | $\zeta_{3} + 1$ |
| $9$ | $6$ | $(1,5,2,9,3,6)$ | $-\zeta_{3}$ |
| $9$ | $6$ | $(1,9)(2,6)(3,5)(4,7,8)$ | $-1$ |
| $9$ | $6$ | $(1,9)(2,6)(3,5)(4,8,7)$ | $-1$ |
| $9$ | $6$ | $(1,4,2,8,3,7)(5,6,9)$ | $-\zeta_{3}$ |
| $9$ | $6$ | $(1,7,3,8,2,4)(5,9,6)$ | $\zeta_{3} + 1$ |
| $18$ | $9$ | $(1,8,9,3,4,5,2,7,6)$ | $0$ |
| $18$ | $9$ | $(1,9,4,2,6,8,3,5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
