Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 107 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 107 }$: $ x^{3} + 5 x + 105 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 42\cdot 107 + 31\cdot 107^{2} + 105\cdot 107^{3} + 16\cdot 107^{4} + 4\cdot 107^{5} + 30\cdot 107^{6} + 50\cdot 107^{7} + 15\cdot 107^{8} + 90\cdot 107^{9} +O\left(107^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 81 + 51\cdot 107 + 5\cdot 107^{2} + 70\cdot 107^{3} + 98\cdot 107^{4} + 38\cdot 107^{5} + 68\cdot 107^{6} + 31\cdot 107^{7} + 66\cdot 107^{8} + 2\cdot 107^{9} +O\left(107^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 104 + 106\cdot 107 + 81\cdot 107^{2} + 28\cdot 107^{3} + 32\cdot 107^{4} + 18\cdot 107^{5} + 67\cdot 107^{6} + 89\cdot 107^{7} + 18\cdot 107^{8} + 21\cdot 107^{9} +O\left(107^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 a^{2} + 67 a + 91 + \left(91 a^{2} + 35 a + 21\right)\cdot 107 + \left(39 a^{2} + 87 a + 63\right)\cdot 107^{2} + \left(31 a^{2} + 45 a + 51\right)\cdot 107^{3} + \left(86 a^{2} + 73 a + 28\right)\cdot 107^{4} + \left(70 a^{2} + 52 a + 10\right)\cdot 107^{5} + \left(51 a^{2} + 19 a + 29\right)\cdot 107^{6} + \left(12 a^{2} + 20 a + 102\right)\cdot 107^{7} + \left(65 a^{2} + 100 a + 63\right)\cdot 107^{8} + \left(52 a^{2} + 23 a + 82\right)\cdot 107^{9} +O\left(107^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 a^{2} + 23 a + 82 + \left(101 a^{2} + 32 a + 88\right)\cdot 107 + \left(45 a^{2} + 42 a + 4\right)\cdot 107^{2} + \left(26 a^{2} + 96 a + 73\right)\cdot 107^{3} + \left(23 a^{2} + 60 a + 37\right)\cdot 107^{4} + \left(97 a^{2} + 97 a + 101\right)\cdot 107^{5} + \left(28 a^{2} + 34 a + 5\right)\cdot 107^{6} + \left(102 a^{2} + 44 a + 9\right)\cdot 107^{7} + \left(89 a^{2} + 16 a + 27\right)\cdot 107^{8} + \left(49 a^{2} + 92 a + 78\right)\cdot 107^{9} +O\left(107^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 a^{2} + 40 a + 102 + \left(38 a^{2} + 97 a + 23\right)\cdot 107 + \left(39 a^{2} + 6 a + 97\right)\cdot 107^{2} + \left(2 a^{2} + 91 a + 61\right)\cdot 107^{3} + \left(85 a^{2} + 34 a + 95\right)\cdot 107^{4} + \left(75 a^{2} + 53 a + 26\right)\cdot 107^{5} + \left(27 a^{2} + 64 a + 56\right)\cdot 107^{6} + \left(45 a^{2} + 48 a + 104\right)\cdot 107^{7} + \left(15 a^{2} + 67 a + 40\right)\cdot 107^{8} + \left(88 a^{2} + 70 a + 58\right)\cdot 107^{9} +O\left(107^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 30 a^{2} + 36 a + 102 + \left(69 a^{2} + 63 a + 17\right)\cdot 107 + \left(71 a^{2} + 56 a + 19\right)\cdot 107^{2} + \left(38 a^{2} + 104 a + 78\right)\cdot 107^{3} + \left(17 a^{2} + 18 a + 53\right)\cdot 107^{4} + \left(87 a^{2} + 16 a + 103\right)\cdot 107^{5} + \left(89 a^{2} + 73 a + 101\right)\cdot 107^{6} + \left(28 a^{2} + 34 a + 13\right)\cdot 107^{7} + \left(61 a^{2} + 46 a + 74\right)\cdot 107^{8} + \left(62 a^{2} + 92 a + 13\right)\cdot 107^{9} +O\left(107^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 53 a^{2} + 48 a + 36 + \left(43 a^{2} + 11 a + 3\right)\cdot 107 + \left(96 a^{2} + 8 a + 66\right)\cdot 107^{2} + \left(41 a^{2} + 13 a + 17\right)\cdot 107^{3} + \left(66 a^{2} + 27 a + 3\right)\cdot 107^{4} + \left(29 a^{2} + 100 a + 90\right)\cdot 107^{5} + \left(95 a^{2} + 105 a + 48\right)\cdot 107^{6} + \left(82 a^{2} + 27 a + 51\right)\cdot 107^{7} + \left(62 a^{2} + 44 a + 43\right)\cdot 107^{8} + \left(101 a^{2} + 29 a + 72\right)\cdot 107^{9} +O\left(107^{ 10 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 63 a^{2} + 37 + \left(84 a^{2} + 81 a + 71\right)\cdot 107 + \left(27 a^{2} + 12 a + 58\right)\cdot 107^{2} + \left(73 a^{2} + 77 a + 48\right)\cdot 107^{3} + \left(42 a^{2} + 105 a + 61\right)\cdot 107^{4} + \left(67 a^{2} + 34\right)\cdot 107^{5} + \left(27 a^{2} + 23 a + 20\right)\cdot 107^{6} + \left(49 a^{2} + 38 a + 82\right)\cdot 107^{7} + \left(26 a^{2} + 46 a + 77\right)\cdot 107^{8} + \left(73 a^{2} + 12 a + 8\right)\cdot 107^{9} +O\left(107^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(4,6,9)(5,8,7)$ |
| $(1,9,8)(2,4,5)(3,6,7)$ |
| $(1,3)(4,6)(5,7)$ |
| $(1,2,3)(4,9,6)$ |
| $(4,5)(6,7)(8,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,3)(4,9)(7,8)$ | $0$ |
| $9$ | $2$ | $(4,5)(6,7)(8,9)$ | $2$ |
| $9$ | $2$ | $(1,4)(2,9)(3,6)(5,7)$ | $0$ |
| $2$ | $3$ | $(1,3,2)(4,9,6)(5,8,7)$ | $-3$ |
| $6$ | $3$ | $(1,2,3)(4,9,6)$ | $0$ |
| $6$ | $3$ | $(1,8,4)(2,5,6)(3,7,9)$ | $0$ |
| $12$ | $3$ | $(1,9,8)(2,4,5)(3,6,7)$ | $0$ |
| $18$ | $6$ | $(1,9,8,3,4,7)(2,6,5)$ | $0$ |
| $18$ | $6$ | $(1,2,3)(4,5,9,8,6,7)$ | $-1$ |
| $18$ | $6$ | $(1,9,3,4,2,6)(5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
