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