Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 139 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 139 }$: $ x^{3} + 6 x + 137 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 23 + 26\cdot 139 + 133\cdot 139^{2} + 97\cdot 139^{3} + 4\cdot 139^{4} + 74\cdot 139^{5} + 62\cdot 139^{6} + 139^{7} + 9\cdot 139^{8} + 29\cdot 139^{9} + 72\cdot 139^{10} +O\left(139^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 52 + 9\cdot 139 + 123\cdot 139^{2} + 85\cdot 139^{3} + 42\cdot 139^{4} + 63\cdot 139^{5} + 3\cdot 139^{6} + 77\cdot 139^{7} + 138\cdot 139^{8} + 66\cdot 139^{9} + 29\cdot 139^{10} +O\left(139^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 64 + 103\cdot 139 + 21\cdot 139^{2} + 94\cdot 139^{3} + 91\cdot 139^{4} + 139^{5} + 73\cdot 139^{6} + 60\cdot 139^{7} + 130\cdot 139^{8} + 42\cdot 139^{9} + 37\cdot 139^{10} +O\left(139^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 53 a^{2} + 62 a + 73 + \left(59 a^{2} + 73 a + 98\right)\cdot 139 + \left(114 a^{2} + 132 a + 40\right)\cdot 139^{2} + \left(47 a^{2} + 53 a + 52\right)\cdot 139^{3} + \left(26 a^{2} + 66 a + 105\right)\cdot 139^{4} + \left(72 a^{2} + 4 a + 10\right)\cdot 139^{5} + \left(17 a^{2} + 90 a + 70\right)\cdot 139^{6} + \left(105 a^{2} + 37 a + 3\right)\cdot 139^{7} + \left(130 a^{2} + 23 a + 106\right)\cdot 139^{8} + \left(25 a^{2} + 90 a + 103\right)\cdot 139^{9} + \left(95 a^{2} + 27 a + 102\right)\cdot 139^{10} +O\left(139^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 62 a^{2} + 109 a + 109 + \left(104 a^{2} + 86 a\right)\cdot 139 + \left(42 a^{2} + 32 a + 32\right)\cdot 139^{2} + \left(89 a^{2} + 4 a + 79\right)\cdot 139^{3} + \left(133 a^{2} + 60 a + 117\right)\cdot 139^{4} + \left(20 a^{2} + 37 a + 83\right)\cdot 139^{5} + \left(a^{2} + 61 a + 4\right)\cdot 139^{6} + \left(50 a^{2} + 128 a + 61\right)\cdot 139^{7} + \left(34 a^{2} + 13 a + 137\right)\cdot 139^{8} + \left(11 a^{2} + 104 a + 44\right)\cdot 139^{9} + \left(83 a^{2} + 7 a + 54\right)\cdot 139^{10} +O\left(139^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 94 a^{2} + 138 a + 98 + \left(31 a^{2} + 49 a + 126\right)\cdot 139 + \left(137 a^{2} + 69 a + 131\right)\cdot 139^{2} + \left(28 a^{2} + 125 a + 115\right)\cdot 139^{3} + \left(74 a^{2} + 136 a + 18\right)\cdot 139^{4} + \left(68 a^{2} + 37 a + 135\right)\cdot 139^{5} + \left(11 a^{2} + 68 a + 45\right)\cdot 139^{6} + \left(130 a^{2} + 45 a + 103\right)\cdot 139^{7} + \left(49 a^{2} + 87 a + 60\right)\cdot 139^{8} + \left(130 a^{2} + 125 a + 104\right)\cdot 139^{9} + \left(100 a^{2} + 84 a + 125\right)\cdot 139^{10} +O\left(139^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 99 a^{2} + 5 a + 118 + \left(77 a^{2} + 4 a + 32\right)\cdot 139 + \left(76 a^{2} + 121 a + 28\right)\cdot 139^{2} + \left(29 a^{2} + 68 a + 118\right)\cdot 139^{3} + \left(70 a^{2} + 134 a + 2\right)\cdot 139^{4} + \left(16 a^{2} + 29 a + 66\right)\cdot 139^{5} + \left(137 a^{2} + 136 a + 131\right)\cdot 139^{6} + \left(105 a^{2} + 66 a + 6\right)\cdot 139^{7} + \left(42 a^{2} + 38 a + 32\right)\cdot 139^{8} + \left(131 a^{2} + 117 a + 108\right)\cdot 139^{9} + \left(65 a^{2} + 53 a + 124\right)\cdot 139^{10} +O\left(139^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 122 a^{2} + 31 a + 71 + \left(2 a^{2} + 2 a + 11\right)\cdot 139 + \left(98 a^{2} + 37 a + 114\right)\cdot 139^{2} + \left(20 a^{2} + 9 a + 82\right)\cdot 139^{3} + \left(70 a^{2} + 81 a + 2\right)\cdot 139^{4} + \left(49 a^{2} + 63 a + 59\right)\cdot 139^{5} + \left(126 a^{2} + 9 a + 88\right)\cdot 139^{6} + \left(97 a^{2} + 104 a + 113\right)\cdot 139^{7} + \left(54 a^{2} + 37 a + 79\right)\cdot 139^{8} + \left(136 a^{2} + 48 a + 128\right)\cdot 139^{9} + \left(93 a^{2} + 46 a + 97\right)\cdot 139^{10} +O\left(139^{ 11 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 126 a^{2} + 72 a + 87 + \left(a^{2} + 61 a + 7\right)\cdot 139 + \left(87 a^{2} + 24 a + 70\right)\cdot 139^{2} + \left(61 a^{2} + 16 a + 107\right)\cdot 139^{3} + \left(42 a^{2} + 77 a + 30\right)\cdot 139^{4} + \left(50 a^{2} + 104 a + 62\right)\cdot 139^{5} + \left(123 a^{2} + 51 a + 76\right)\cdot 139^{6} + \left(66 a^{2} + 34 a + 128\right)\cdot 139^{7} + \left(104 a^{2} + 77 a\right)\cdot 139^{8} + \left(120 a^{2} + 70 a + 66\right)\cdot 139^{9} + \left(116 a^{2} + 57 a + 50\right)\cdot 139^{10} +O\left(139^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3,2)(4,7,9)(5,8,6)$ |
| $(1,9)(2,4)(3,7)(5,6)$ |
| $(1,2,3)(4,7,9)$ |
| $(1,4,5)(2,9,6)(3,7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,9)(2,4)(3,7)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,3,2)(4,7,9)(5,8,6)$ | $-3$ |
| $3$ | $3$ | $(1,2,3)(5,8,6)$ | $0$ |
| $3$ | $3$ | $(1,3,2)(5,6,8)$ | $0$ |
| $6$ | $3$ | $(1,4,5)(2,9,6)(3,7,8)$ | $0$ |
| $6$ | $3$ | $(1,7,8)(2,4,5)(3,9,6)$ | $0$ |
| $6$ | $3$ | $(1,8,7)(2,5,4)(3,6,9)$ | $0$ |
| $9$ | $6$ | $(1,6,2,5,3,8)(7,9)$ | $0$ |
| $9$ | $6$ | $(1,8,3,5,2,6)(7,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
