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