Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{3} + 5 x + 57 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 23 a^{2} + 2 a + 3 + \left(18 a^{2} + 10 a + 13\right)\cdot 59 + \left(29 a^{2} + 36 a + 53\right)\cdot 59^{2} + \left(38 a^{2} + 51 a + 33\right)\cdot 59^{3} + \left(30 a^{2} + 47 a + 13\right)\cdot 59^{4} + \left(51 a^{2} + 29 a + 28\right)\cdot 59^{5} + \left(21 a^{2} + 30 a + 46\right)\cdot 59^{6} + \left(4 a^{2} + 43 a + 39\right)\cdot 59^{7} + \left(46 a^{2} + 55 a + 16\right)\cdot 59^{8} + \left(35 a^{2} + 31 a + 33\right)\cdot 59^{9} + \left(22 a^{2} + 50 a + 39\right)\cdot 59^{10} + \left(32 a^{2} + 2 a + 11\right)\cdot 59^{11} +O\left(59^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 a^{2} + 17 a + 6 + \left(20 a^{2} + 26 a + 39\right)\cdot 59 + \left(30 a^{2} + 51 a + 56\right)\cdot 59^{2} + \left(50 a^{2} + 33 a + 14\right)\cdot 59^{3} + \left(16 a^{2} + 35 a + 46\right)\cdot 59^{4} + \left(38 a^{2} + 47 a + 3\right)\cdot 59^{5} + \left(7 a^{2} + 56 a + 58\right)\cdot 59^{6} + \left(55 a^{2} + 13 a + 31\right)\cdot 59^{7} + \left(8 a^{2} + 45 a + 30\right)\cdot 59^{8} + \left(16 a^{2} + 56 a + 46\right)\cdot 59^{9} + \left(18 a^{2} + 13 a + 44\right)\cdot 59^{10} + \left(44 a^{2} + 33 a + 31\right)\cdot 59^{11} +O\left(59^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 46 a^{2} + 40 a + 42 + \left(30 a^{2} + 2 a + 3\right)\cdot 59 + \left(47 a^{2} + 20 a + 54\right)\cdot 59^{2} + \left(54 a^{2} + 56 a + 18\right)\cdot 59^{3} + \left(45 a^{2} + 21 a + 34\right)\cdot 59^{4} + \left(2 a^{2} + 5 a + 8\right)\cdot 59^{5} + \left(27 a^{2} + 25 a + 30\right)\cdot 59^{6} + \left(25 a^{2} + a + 2\right)\cdot 59^{7} + \left(15 a^{2} + 19 a + 38\right)\cdot 59^{8} + \left(8 a^{2} + 54 a + 7\right)\cdot 59^{9} + \left(6 a^{2} + 3 a + 51\right)\cdot 59^{10} + \left(10 a^{2} + 55 a + 48\right)\cdot 59^{11} +O\left(59^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 + 25\cdot 59 + 39\cdot 59^{2} + 43\cdot 59^{3} + 49\cdot 59^{4} + 38\cdot 59^{5} + 47\cdot 59^{6} + 37\cdot 59^{7} + 46\cdot 59^{8} + 44\cdot 59^{9} + 56\cdot 59^{10} + 29\cdot 59^{11} +O\left(59^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 53 + 13\cdot 59 + 22\cdot 59^{2} + 52\cdot 59^{3} + 14\cdot 59^{4} + 43\cdot 59^{5} + 44\cdot 59^{6} + 58\cdot 59^{7} + 22\cdot 59^{8} + 4\cdot 59^{9} + 16\cdot 59^{10} + 9\cdot 59^{11} +O\left(59^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 31 a^{2} + 13 a + 51 + \left(13 a^{2} + 18 a + 24\right)\cdot 59 + \left(31 a^{2} + 14 a + 19\right)\cdot 59^{2} + \left(22 a^{2} + 55 a + 29\right)\cdot 59^{3} + \left(22 a^{2} + 48 a + 34\right)\cdot 59^{4} + \left(14 a^{2} + 20 a + 27\right)\cdot 59^{5} + \left(7 a^{2} + 30 a + 3\right)\cdot 59^{6} + \left(33 a^{2} + 40 a + 28\right)\cdot 59^{7} + \left(18 a^{2} + 35 a + 48\right)\cdot 59^{8} + \left(11 a^{2} + 48 a + 17\right)\cdot 59^{9} + \left(25 a^{2} + 20 a + 16\right)\cdot 59^{10} + \left(12 a^{2} + 25 a + 17\right)\cdot 59^{11} +O\left(59^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 18 a^{2} + 40 a + 6 + \left(20 a^{2} + 22 a + 39\right)\cdot 59 + \left(58 a^{2} + 30 a + 51\right)\cdot 59^{2} + \left(28 a^{2} + 32 a + 21\right)\cdot 59^{3} + \left(11 a^{2} + 34 a + 28\right)\cdot 59^{4} + \left(28 a^{2} + 40 a + 9\right)\cdot 59^{5} + \left(29 a^{2} + 30 a + 52\right)\cdot 59^{6} + \left(58 a^{2} + a + 3\right)\cdot 59^{7} + \left(3 a^{2} + 17 a + 14\right)\cdot 59^{8} + \left(7 a^{2} + 29 a + 16\right)\cdot 59^{9} + \left(18 a^{2} + 53 a + 44\right)\cdot 59^{10} + \left(41 a^{2} + 22 a + 21\right)\cdot 59^{11} +O\left(59^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 41 a^{2} + 6 a + 45 + \left(14 a^{2} + 38 a + 28\right)\cdot 59 + \left(39 a^{2} + 24 a + 26\right)\cdot 59^{2} + \left(40 a^{2} + 6 a + 50\right)\cdot 59^{3} + \left(49 a^{2} + 47 a + 46\right)\cdot 59^{4} + \left(41 a^{2} + 32 a + 20\right)\cdot 59^{5} + \left(24 a^{2} + 3 a + 22\right)\cdot 59^{6} + \left(17 a + 17\right)\cdot 59^{7} + \left(25 a^{2} + 4 a + 50\right)\cdot 59^{8} + \left(39 a^{2} + 15 a + 32\right)\cdot 59^{9} + \left(27 a^{2} + 34 a + 24\right)\cdot 59^{10} + \left(36 a^{2} + 37 a + 38\right)\cdot 59^{11} +O\left(59^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 3 + 48\cdot 59 + 30\cdot 59^{2} + 29\cdot 59^{3} + 26\cdot 59^{4} + 55\cdot 59^{5} + 48\cdot 59^{6} + 15\cdot 59^{7} + 27\cdot 59^{8} + 32\cdot 59^{9} + 59^{10} + 27\cdot 59^{11} +O\left(59^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(2,7)(3,9)(4,8)(5,6)$ |
| $(1,7,2)(4,9,5)$ |
| $(1,5,6,2,4,3,7,9,8)$ |
| $(3,8,6)(4,5,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,3)(2,6)(4,9)(7,8)$ | $0$ |
| $2$ | $3$ | $(1,2,7)(3,8,6)(4,9,5)$ | $-3$ |
| $3$ | $3$ | $(1,7,2)(4,9,5)$ | $0$ |
| $3$ | $3$ | $(1,2,7)(4,5,9)$ | $0$ |
| $9$ | $6$ | $(1,6,7,3,2,8)(4,9)$ | $0$ |
| $9$ | $6$ | $(1,8,2,3,7,6)(4,9)$ | $0$ |
| $6$ | $9$ | $(1,5,6,2,4,3,7,9,8)$ | $0$ |
| $6$ | $9$ | $(1,6,4,7,8,5,2,3,9)$ | $0$ |
| $6$ | $9$ | $(1,3,4,7,6,5,2,8,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
