Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 14.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{3} + 5 x + 57 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 33 a^{2} + 12 a + 30 + \left(40 a^{2} + 18 a + 11\right)\cdot 59 + \left(12 a^{2} + 47 a + 45\right)\cdot 59^{2} + \left(12 a^{2} + 21 a + 49\right)\cdot 59^{3} + \left(17 a^{2} + 47 a + 28\right)\cdot 59^{4} + \left(21 a^{2} + 29 a + 32\right)\cdot 59^{5} + \left(51 a^{2} + 18 a + 41\right)\cdot 59^{6} + \left(51 a^{2} + 21 a + 56\right)\cdot 59^{7} + \left(43 a^{2} + 50 a + 41\right)\cdot 59^{8} + \left(18 a^{2} + 6 a + 39\right)\cdot 59^{9} + \left(50 a^{2} + 9 a + 55\right)\cdot 59^{10} + \left(29 a^{2} + 44 a + 11\right)\cdot 59^{11} + \left(57 a^{2} + 55 a + 48\right)\cdot 59^{12} + \left(11 a^{2} + 17 a + 28\right)\cdot 59^{13} +O\left(59^{ 14 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 48 a^{2} + 48 a + 16 + \left(6 a^{2} + 56 a + 27\right)\cdot 59 + \left(9 a^{2} + 57 a + 24\right)\cdot 59^{2} + \left(49 a^{2} + 4 a + 6\right)\cdot 59^{3} + \left(a^{2} + 32 a + 46\right)\cdot 59^{4} + \left(57 a^{2} + 37 a + 17\right)\cdot 59^{5} + \left(48 a^{2} + 10 a\right)\cdot 59^{6} + \left(20 a^{2} + 23 a + 48\right)\cdot 59^{7} + \left(10 a^{2} + 17 a + 23\right)\cdot 59^{8} + \left(20 a^{2} + 20 a + 51\right)\cdot 59^{9} + \left(41 a^{2} + 2 a + 32\right)\cdot 59^{10} + \left(26 a^{2} + 28 a + 3\right)\cdot 59^{11} + \left(35 a^{2} + 46 a + 9\right)\cdot 59^{12} + \left(16 a^{2} + 51 a + 50\right)\cdot 59^{13} +O\left(59^{ 14 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 a^{2} + 46 a + 3 + \left(4 a^{2} + 15 a + 48\right)\cdot 59 + \left(53 a^{2} + 50 a + 2\right)\cdot 59^{2} + \left(7 a^{2} + 18 a + 55\right)\cdot 59^{3} + \left(11 a^{2} + 42 a + 47\right)\cdot 59^{4} + \left(38 a^{2} + 32 a + 29\right)\cdot 59^{5} + \left(48 a^{2} + a + 32\right)\cdot 59^{6} + \left(44 a^{2} + 36 a + 13\right)\cdot 59^{7} + \left(6 a^{2} + 12 a + 36\right)\cdot 59^{8} + \left(34 a^{2} + 12 a + 51\right)\cdot 59^{9} + \left(3 a^{2} + 7 a + 17\right)\cdot 59^{10} + \left(34 a^{2} + 24 a + 6\right)\cdot 59^{11} + \left(13 a^{2} + 56 a\right)\cdot 59^{12} + \left(13 a^{2} + 14 a + 33\right)\cdot 59^{13} +O\left(59^{ 14 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 + 4\cdot 59 + 17\cdot 59^{2} + 37\cdot 59^{3} + 46\cdot 59^{4} + 16\cdot 59^{5} + 37\cdot 59^{6} + 49\cdot 59^{7} + 7\cdot 59^{8} + 4\cdot 59^{9} + 34\cdot 59^{10} + 53\cdot 59^{11} + 25\cdot 59^{12} +O\left(59^{ 14 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 a^{2} + a + 22 + \left(14 a^{2} + 25 a + 41\right)\cdot 59 + \left(52 a^{2} + 20 a + 19\right)\cdot 59^{2} + \left(38 a^{2} + 18 a + 40\right)\cdot 59^{3} + \left(30 a^{2} + 28 a + 14\right)\cdot 59^{4} + \left(58 a^{2} + 55 a + 58\right)\cdot 59^{5} + \left(17 a^{2} + 38 a + 8\right)\cdot 59^{6} + \left(21 a^{2} + a + 53\right)\cdot 59^{7} + \left(8 a^{2} + 55 a + 21\right)\cdot 59^{8} + \left(6 a^{2} + 39 a + 17\right)\cdot 59^{9} + \left(5 a^{2} + 42 a + 3\right)\cdot 59^{10} + \left(54 a^{2} + 49 a + 14\right)\cdot 59^{11} + \left(46 a^{2} + 5 a + 52\right)\cdot 59^{12} + \left(33 a^{2} + 26 a + 22\right)\cdot 59^{13} +O\left(59^{ 14 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 54 + 26\cdot 59 + 39\cdot 59^{2} + 40\cdot 59^{3} + 3\cdot 59^{4} + 6\cdot 59^{5} + 46\cdot 59^{6} + 57\cdot 59^{7} + 25\cdot 59^{8} + 36\cdot 59^{9} + 23\cdot 59^{10} + 27\cdot 59^{11} + 11\cdot 59^{12} + 29\cdot 59^{13} +O\left(59^{ 14 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 18 a^{2} + 37 a + 34 + \left(43 a^{2} + 32 a + 50\right)\cdot 59 + \left(15 a^{2} + 7 a + 26\right)\cdot 59^{2} + \left(6 a^{2} + 14 a + 40\right)\cdot 59^{3} + \left(12 a^{2} + 53 a + 1\right)\cdot 59^{4} + \left(33 a^{2} + 3 a + 17\right)\cdot 59^{5} + \left(18 a^{2} + 26 a + 56\right)\cdot 59^{6} + \left(58 a^{2} + 29 a + 54\right)\cdot 59^{7} + \left(45 a^{2} + 34 a + 4\right)\cdot 59^{8} + \left(30 a^{2} + 8\right)\cdot 59^{9} + \left(40 a + 54\right)\cdot 59^{10} + \left(14 a^{2} + 50 a + 39\right)\cdot 59^{11} + \left(41 a^{2} + 40 a + 8\right)\cdot 59^{12} + \left(44 a^{2} + 6 a + 6\right)\cdot 59^{13} +O\left(59^{ 14 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 45 + 30\cdot 59 + 11\cdot 59^{2} + 13\cdot 59^{3} + 33\cdot 59^{4} + 19\cdot 59^{5} + 27\cdot 59^{6} + 10\cdot 59^{7} + 16\cdot 59^{8} + 16\cdot 59^{9} + 3\cdot 59^{10} + 25\cdot 59^{11} + 12\cdot 59^{12} + 19\cdot 59^{13} +O\left(59^{ 14 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 52 a^{2} + 33 a + 49 + \left(8 a^{2} + 28 a + 53\right)\cdot 59 + \left(34 a^{2} + 52 a + 48\right)\cdot 59^{2} + \left(3 a^{2} + 39 a + 11\right)\cdot 59^{3} + \left(45 a^{2} + 32 a + 13\right)\cdot 59^{4} + \left(27 a^{2} + 17 a + 38\right)\cdot 59^{5} + \left(50 a^{2} + 22 a + 44\right)\cdot 59^{6} + \left(38 a^{2} + 6 a + 9\right)\cdot 59^{7} + \left(2 a^{2} + 7 a + 57\right)\cdot 59^{8} + \left(8 a^{2} + 38 a + 10\right)\cdot 59^{9} + \left(17 a^{2} + 16 a + 11\right)\cdot 59^{10} + \left(18 a^{2} + 39 a + 54\right)\cdot 59^{11} + \left(41 a^{2} + 30 a + 8\right)\cdot 59^{12} + \left(56 a^{2} + 46\right)\cdot 59^{13} +O\left(59^{ 14 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,6,2)(3,4,7)(5,8,9)$ |
| $(1,4)(2,7)(3,6)(5,8)$ |
| $(1,5,3)(2,9,7)(4,6,8)$ |
| $(1,3,5)(4,6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $0$ |
| $2$ | $3$ | $(1,5,3)(2,9,7)(4,6,8)$ | $-3$ |
| $3$ | $3$ | $(1,3,5)(2,9,7)$ | $0$ |
| $3$ | $3$ | $(1,5,3)(2,7,9)$ | $0$ |
| $6$ | $3$ | $(1,6,2)(3,4,7)(5,8,9)$ | $0$ |
| $6$ | $3$ | $(1,8,9)(2,3,6)(4,7,5)$ | $0$ |
| $6$ | $3$ | $(1,9,8)(2,6,3)(4,5,7)$ | $0$ |
| $9$ | $6$ | $(1,7,3,2,5,9)(4,8)$ | $0$ |
| $9$ | $6$ | $(1,9,5,2,3,7)(4,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
