Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 107 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 107 }$: $ x^{3} + 5 x + 105 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 55 + 83\cdot 107 + 92\cdot 107^{2} + 95\cdot 107^{3} + 99\cdot 107^{4} + 35\cdot 107^{5} + 26\cdot 107^{6} + 83\cdot 107^{7} + 93\cdot 107^{8} + 87\cdot 107^{9} + 80\cdot 107^{10} +O\left(107^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 71 + 68\cdot 107 + 9\cdot 107^{2} + 11\cdot 107^{3} + 43\cdot 107^{4} + 96\cdot 107^{5} + 25\cdot 107^{6} + 54\cdot 107^{7} + 104\cdot 107^{8} + 85\cdot 107^{9} + 51\cdot 107^{10} +O\left(107^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 93 + 64\cdot 107 + 94\cdot 107^{2} + 61\cdot 107^{3} + 58\cdot 107^{4} + 55\cdot 107^{5} + 104\cdot 107^{6} + 43\cdot 107^{7} + 53\cdot 107^{8} + 97\cdot 107^{9} + 73\cdot 107^{10} +O\left(107^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a^{2} + 41 a + 64 + \left(31 a^{2} + 68 a + 35\right)\cdot 107 + \left(53 a^{2} + 106 a + 72\right)\cdot 107^{2} + \left(64 a^{2} + 8 a + 90\right)\cdot 107^{3} + \left(32 a^{2} + 66 a + 27\right)\cdot 107^{4} + \left(60 a^{2} + 84 a + 82\right)\cdot 107^{5} + \left(98 a^{2} + 59 a + 42\right)\cdot 107^{6} + \left(94 a^{2} + 14 a + 20\right)\cdot 107^{7} + \left(87 a^{2} + 11 a + 104\right)\cdot 107^{8} + \left(49 a^{2} + 25 a + 1\right)\cdot 107^{9} + \left(94 a^{2} + 80 a + 83\right)\cdot 107^{10} +O\left(107^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 13 a^{2} + 77 a + 1 + \left(33 a^{2} + 5 a + 106\right)\cdot 107 + \left(31 a^{2} + 78 a + 36\right)\cdot 107^{2} + \left(73 a^{2} + 25 a + 98\right)\cdot 107^{3} + \left(26 a^{2} + 80 a + 66\right)\cdot 107^{4} + \left(86 a^{2} + 83 a + 22\right)\cdot 107^{5} + \left(106 a^{2} + 14 a + 90\right)\cdot 107^{6} + \left(15 a^{2} + 59 a + 74\right)\cdot 107^{7} + \left(96 a^{2} + 2 a + 68\right)\cdot 107^{8} + \left(72 a^{2} + 6 a + 102\right)\cdot 107^{9} + \left(90 a^{2} + 41 a + 72\right)\cdot 107^{10} +O\left(107^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 a^{2} + 45 a + 57 + \left(93 a^{2} + 74 a + 100\right)\cdot 107 + \left(92 a^{2} + 82 a + 25\right)\cdot 107^{2} + \left(79 a^{2} + 33 a + 106\right)\cdot 107^{3} + \left(70 a^{2} + 10 a + 47\right)\cdot 107^{4} + \left(58 a^{2} + 105 a + 5\right)\cdot 107^{5} + \left(14 a^{2} + 57 a + 48\right)\cdot 107^{6} + \left(92 a^{2} + 4 a + 82\right)\cdot 107^{7} + \left(35 a^{2} + 18 a + 37\right)\cdot 107^{8} + \left(30 a^{2} + 39 a + 8\right)\cdot 107^{9} + \left(42 a^{2} + 61 a + 16\right)\cdot 107^{10} +O\left(107^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 40 a^{2} + 17 a + 91 + \left(52 a^{2} + 74 a + 98\right)\cdot 107 + \left(55 a^{2} + 17 a + 81\right)\cdot 107^{2} + \left(35 a^{2} + 10 a + 43\right)\cdot 107^{3} + \left(52 a^{2} + 16 a + 45\right)\cdot 107^{4} + \left(84 a^{2} + 87 a + 52\right)\cdot 107^{5} + \left(25 a + 93\right)\cdot 107^{6} + \left(30 a^{2} + 35 a + 85\right)\cdot 107^{7} + \left(93 a^{2} + 86 a + 94\right)\cdot 107^{8} + \left(61 a^{2} + 92 a + 65\right)\cdot 107^{9} + \left(95 a^{2} + 98 a + 53\right)\cdot 107^{10} +O\left(107^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 54 a^{2} + 13 a + 102 + \left(21 a^{2} + 27 a + 102\right)\cdot 107 + \left(20 a^{2} + 11 a + 106\right)\cdot 107^{2} + \left(105 a^{2} + 71 a + 61\right)\cdot 107^{3} + \left(27 a^{2} + 10 a + 35\right)\cdot 107^{4} + \left(43 a^{2} + 43 a + 93\right)\cdot 107^{5} + \left(106 a^{2} + 66 a + 88\right)\cdot 107^{6} + \left(60 a^{2} + 12 a + 10\right)\cdot 107^{7} + \left(24 a^{2} + 18 a + 44\right)\cdot 107^{8} + \left(79 a^{2} + 8 a + 52\right)\cdot 107^{9} + \left(27 a^{2} + 74 a + 41\right)\cdot 107^{10} +O\left(107^{ 11 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 63 a^{2} + 21 a + 1 + \left(89 a^{2} + 71 a + 88\right)\cdot 107 + \left(67 a^{2} + 24 a + 13\right)\cdot 107^{2} + \left(69 a^{2} + 64 a + 72\right)\cdot 107^{3} + \left(3 a^{2} + 30 a + 2\right)\cdot 107^{4} + \left(95 a^{2} + 24 a + 91\right)\cdot 107^{5} + \left(100 a^{2} + 96 a + 14\right)\cdot 107^{6} + \left(26 a^{2} + 87 a + 79\right)\cdot 107^{7} + \left(90 a^{2} + 77 a + 40\right)\cdot 107^{8} + \left(26 a^{2} + 42 a + 32\right)\cdot 107^{9} + \left(77 a^{2} + 72 a + 61\right)\cdot 107^{10} +O\left(107^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,2,3)(4,6,9)$ |
| $(1,2,3)(5,8,7)$ |
| $(1,8,9)(2,5,6)(3,7,4)$ |
| $(2,3)(4,6)(5,7)$ |
| $(1,9)(2,6)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(2,3)(4,6)(5,7)$ | $0$ |
| $9$ | $2$ | $(1,9)(2,6)(3,4)$ | $2$ |
| $9$ | $2$ | $(1,9)(2,4)(3,6)(5,7)$ | $0$ |
| $2$ | $3$ | $(1,2,3)(4,9,6)(5,7,8)$ | $-3$ |
| $6$ | $3$ | $(1,8,9)(2,5,6)(3,7,4)$ | $0$ |
| $6$ | $3$ | $(1,2,3)(5,8,7)$ | $0$ |
| $12$ | $3$ | $(1,8,4)(2,5,9)(3,7,6)$ | $0$ |
| $18$ | $6$ | $(1,8,9)(2,7,6,3,5,4)$ | $0$ |
| $18$ | $6$ | $(1,9,2,6,3,4)(5,8,7)$ | $-1$ |
| $18$ | $6$ | $(1,9,2,4,3,6)(7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
