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