Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 137 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 137 }$: $ x^{3} + 6 x + 134 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 37\cdot 137 + 73\cdot 137^{2} + 87\cdot 137^{3} + 95\cdot 137^{4} + 59\cdot 137^{5} + 27\cdot 137^{6} + 29\cdot 137^{7} + 17\cdot 137^{8} + 91\cdot 137^{9} +O\left(137^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 71 + 8\cdot 137 + 7\cdot 137^{2} + 93\cdot 137^{3} + 11\cdot 137^{4} + 114\cdot 137^{5} + 79\cdot 137^{6} + 27\cdot 137^{7} + 91\cdot 137^{8} + 132\cdot 137^{9} +O\left(137^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 115 + 34\cdot 137 + 94\cdot 137^{2} + 34\cdot 137^{3} + 21\cdot 137^{4} + 33\cdot 137^{5} + 48\cdot 137^{6} + 110\cdot 137^{7} + 134\cdot 137^{8} + 30\cdot 137^{9} +O\left(137^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 a^{2} + 3 a + 26 + \left(122 a^{2} + 120 a + 100\right)\cdot 137 + \left(114 a^{2} + 25 a + 117\right)\cdot 137^{2} + \left(50 a^{2} + 60 a + 74\right)\cdot 137^{3} + \left(41 a^{2} + 10 a + 51\right)\cdot 137^{4} + \left(33 a^{2} + a + 119\right)\cdot 137^{5} + \left(126 a^{2} + 123 a + 21\right)\cdot 137^{6} + \left(72 a^{2} + 67 a + 36\right)\cdot 137^{7} + \left(50 a^{2} + 17 a + 117\right)\cdot 137^{8} + \left(52 a^{2} + 23 a + 30\right)\cdot 137^{9} +O\left(137^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 86 a^{2} + 45 a + 40 + \left(46 a^{2} + 50 a\right)\cdot 137 + \left(127 a^{2} + 57 a + 108\right)\cdot 137^{2} + \left(6 a^{2} + 22 a + 38\right)\cdot 137^{3} + \left(100 a^{2} + 76 a + 60\right)\cdot 137^{4} + \left(116 a^{2} + 116 a + 46\right)\cdot 137^{5} + \left(49 a^{2} + 8 a + 128\right)\cdot 137^{6} + \left(66 a^{2} + 72 a + 99\right)\cdot 137^{7} + \left(101 a^{2} + 136 a + 135\right)\cdot 137^{8} + \left(67 a^{2} + 7 a + 44\right)\cdot 137^{9} +O\left(137^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 91 a^{2} + 104 a + 60 + \left(96 a^{2} + 29 a + 63\right)\cdot 137 + \left(123 a^{2} + 9 a + 93\right)\cdot 137^{2} + \left(97 a^{2} + 53 a + 128\right)\cdot 137^{3} + \left(48 a^{2} + 51 a + 128\right)\cdot 137^{4} + \left(4 a^{2} + 34 a + 7\right)\cdot 137^{5} + \left(86 a^{2} + 76 a + 136\right)\cdot 137^{6} + \left(17 a^{2} + 24 a + 41\right)\cdot 137^{7} + \left(70 a^{2} + 98 a + 10\right)\cdot 137^{8} + \left(121 a^{2} + 128 a + 123\right)\cdot 137^{9} +O\left(137^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 97 a^{2} + 125 a + 84 + \left(130 a^{2} + 56 a + 62\right)\cdot 137 + \left(22 a^{2} + 70 a + 101\right)\cdot 137^{2} + \left(32 a^{2} + 61 a + 2\right)\cdot 137^{3} + \left(125 a^{2} + 9 a + 24\right)\cdot 137^{4} + \left(15 a^{2} + 123 a + 54\right)\cdot 137^{5} + \left(a^{2} + 51 a + 70\right)\cdot 137^{6} + \left(53 a^{2} + 40 a + 46\right)\cdot 137^{7} + \left(102 a^{2} + 39 a + 2\right)\cdot 137^{8} + \left(84 a^{2} + 113\right)\cdot 137^{9} +O\left(137^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 116 a^{2} + 85 a + 112 + \left(97 a^{2} + 40 a + 2\right)\cdot 137 + \left(26 a^{2} + 26 a + 39\right)\cdot 137^{2} + \left(105 a^{2} + 112 a + 18\right)\cdot 137^{3} + \left(85 a^{2} + 24 a + 92\right)\cdot 137^{4} + \left(106 a^{2} + 109 a + 1\right)\cdot 137^{5} + \left(124 a^{2} + 78 a + 16\right)\cdot 137^{6} + \left(113 a^{2} + 56 a + 63\right)\cdot 137^{7} + \left(37 a^{2} + 135 a + 66\right)\cdot 137^{8} + \left(94 a^{2} + 6 a + 61\right)\cdot 137^{9} +O\left(137^{ 10 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 132 a^{2} + 49 a + 39 + \left(53 a^{2} + 113 a + 101\right)\cdot 137 + \left(132 a^{2} + 84 a + 50\right)\cdot 137^{2} + \left(117 a^{2} + 101 a + 69\right)\cdot 137^{3} + \left(9 a^{2} + 101 a + 62\right)\cdot 137^{4} + \left(134 a^{2} + 26 a + 111\right)\cdot 137^{5} + \left(22 a^{2} + 72 a + 19\right)\cdot 137^{6} + \left(87 a^{2} + 12 a + 93\right)\cdot 137^{7} + \left(48 a^{2} + 121 a + 109\right)\cdot 137^{8} + \left(127 a^{2} + 106 a + 56\right)\cdot 137^{9} +O\left(137^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3,2)(4,9,8)(5,7,6)$ |
| $(4,9,8)(5,6,7)$ |
| $(1,6,4)(2,7,8)(3,5,9)$ |
| $(4,5)(6,8)(7,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $9$ | $2$ | $(1,9)(2,4)(3,8)$ | $-1$ |
| $1$ | $3$ | $(1,3,2)(4,9,8)(5,7,6)$ | $3 \zeta_{3}$ |
| $1$ | $3$ | $(1,2,3)(4,8,9)(5,6,7)$ | $-3 \zeta_{3} - 3$ |
| $6$ | $3$ | $(1,6,4)(2,7,8)(3,5,9)$ | $0$ |
| $6$ | $3$ | $(1,7,4)(2,5,8)(3,6,9)$ | $0$ |
| $6$ | $3$ | $(4,9,8)(5,6,7)$ | $0$ |
| $6$ | $3$ | $(1,4,5)(2,8,6)(3,9,7)$ | $0$ |
| $9$ | $6$ | $(1,8,2,9,3,4)(5,7,6)$ | $-\zeta_{3}$ |
| $9$ | $6$ | $(1,4,3,9,2,8)(5,6,7)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
