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