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