Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 15.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{3} + 5 x + 57 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 54 a^{2} + 56 a + 9 + \left(45 a^{2} + 48 a + 23\right)\cdot 59 + \left(8 a^{2} + 53 a + 58\right)\cdot 59^{2} + \left(32 a^{2} + 43 a + 26\right)\cdot 59^{3} + \left(45 a^{2} + 9 a + 41\right)\cdot 59^{4} + \left(48 a^{2} + 33 a + 48\right)\cdot 59^{5} + \left(37 a^{2} + 27 a + 42\right)\cdot 59^{6} + \left(47 a^{2} + 46 a + 12\right)\cdot 59^{7} + \left(43 a^{2} + 39 a + 4\right)\cdot 59^{8} + \left(33 a^{2} + 20 a + 39\right)\cdot 59^{9} + \left(49 a^{2} + 12 a + 57\right)\cdot 59^{10} + \left(43 a^{2} + 42 a + 26\right)\cdot 59^{11} + \left(33 a^{2} + 34 a + 11\right)\cdot 59^{12} + \left(39 a^{2} + 9 a + 19\right)\cdot 59^{13} + \left(20 a^{2} + 7 a + 23\right)\cdot 59^{14} +O\left(59^{ 15 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 a^{2} + 58 a + 34 + \left(44 a^{2} + 25 a + 18\right)\cdot 59 + \left(43 a^{2} + 37\right)\cdot 59^{2} + \left(18 a^{2} + 40 a + 21\right)\cdot 59^{3} + \left(8 a^{2} + 45 a + 35\right)\cdot 59^{4} + \left(8 a^{2} + 34 a + 11\right)\cdot 59^{5} + \left(16 a^{2} + 35 a + 49\right)\cdot 59^{6} + \left(33 a^{2} + 22 a + 23\right)\cdot 59^{7} + \left(23 a^{2} + 9 a + 15\right)\cdot 59^{8} + \left(56 a^{2} + a + 16\right)\cdot 59^{9} + \left(26 a^{2} + 44 a + 41\right)\cdot 59^{10} + \left(35 a^{2} + 30 a + 18\right)\cdot 59^{11} + \left(4 a^{2} + 18 a + 32\right)\cdot 59^{12} + \left(10 a^{2} + 36 a + 58\right)\cdot 59^{13} + \left(52 a^{2} + 14 a + 29\right)\cdot 59^{14} +O\left(59^{ 15 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 10\cdot 59 + 10\cdot 59^{2} + 21\cdot 59^{3} + 46\cdot 59^{4} + 51\cdot 59^{5} + 53\cdot 59^{6} + 11\cdot 59^{7} + 34\cdot 59^{8} + 48\cdot 59^{9} + 27\cdot 59^{10} + 8\cdot 59^{11} + 50\cdot 59^{12} + 44\cdot 59^{13} + 49\cdot 59^{14} +O\left(59^{ 15 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 32 a^{2} + 4 a + 34 + \left(27 a^{2} + 43 a + 40\right)\cdot 59 + \left(6 a^{2} + 4 a + 50\right)\cdot 59^{2} + \left(8 a^{2} + 34 a + 5\right)\cdot 59^{3} + \left(5 a^{2} + 3 a + 5\right)\cdot 59^{4} + \left(2 a^{2} + 50 a + 11\right)\cdot 59^{5} + \left(5 a^{2} + 54 a + 12\right)\cdot 59^{6} + \left(37 a^{2} + 48 a + 56\right)\cdot 59^{7} + \left(50 a^{2} + 9 a + 26\right)\cdot 59^{8} + \left(27 a^{2} + 37 a + 19\right)\cdot 59^{9} + \left(41 a^{2} + 2 a + 50\right)\cdot 59^{10} + \left(38 a^{2} + 45 a + 9\right)\cdot 59^{11} + \left(20 a^{2} + 5 a + 7\right)\cdot 59^{12} + \left(9 a^{2} + 13 a + 56\right)\cdot 59^{13} + \left(45 a^{2} + 37 a + 45\right)\cdot 59^{14} +O\left(59^{ 15 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 23 + 9\cdot 59 + 18\cdot 59^{2} + 43\cdot 59^{3} + 42\cdot 59^{4} + 31\cdot 59^{5} + 16\cdot 59^{6} + 6\cdot 59^{7} + 30\cdot 59^{8} + 21\cdot 59^{9} + 14\cdot 59^{10} + 19\cdot 59^{11} + 23\cdot 59^{12} + 30\cdot 59^{13} + 3\cdot 59^{14} +O\left(59^{ 15 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 8 a^{2} + 53 a + 3 + \left(34 a^{2} + 25 a + 29\right)\cdot 59 + \left(7 a^{2} + 44 a + 50\right)\cdot 59^{2} + \left(16 a^{2} + a + 29\right)\cdot 59^{3} + \left(58 a^{2} + 47 a + 30\right)\cdot 59^{4} + \left(16 a^{2} + 4 a + 52\right)\cdot 59^{5} + \left(46 a^{2} + 56 a + 25\right)\cdot 59^{6} + \left(34 a^{2} + 3 a + 43\right)\cdot 59^{7} + \left(51 a^{2} + 56 a + 27\right)\cdot 59^{8} + \left(33 a^{2} + 7 a + 20\right)\cdot 59^{9} + \left(21 a^{2} + 21\right)\cdot 59^{10} + \left(26 a^{2} + 18 a + 56\right)\cdot 59^{11} + \left(54 a^{2} + 35 a + 55\right)\cdot 59^{12} + \left(55 a^{2} + 20 a + 54\right)\cdot 59^{13} + \left(56 a^{2} + 32 a + 1\right)\cdot 59^{14} +O\left(59^{ 15 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 27 a^{2} + 53 a + 27 + \left(10 a^{2} + 44 a + 48\right)\cdot 59 + \left(58 a^{2} + 12 a + 2\right)\cdot 59^{2} + \left(49 a^{2} + 15 a + 5\right)\cdot 59^{3} + \left(2 a^{2} + 36 a + 3\right)\cdot 59^{4} + \left(41 a^{2} + 3 a + 54\right)\cdot 59^{5} + \left(36 a^{2} + 39 a + 52\right)\cdot 59^{6} + \left(39 a^{2} + 47 a + 39\right)\cdot 59^{7} + \left(21 a^{2} + 44 a + 6\right)\cdot 59^{8} + \left(24 a^{2} + 30 a + 28\right)\cdot 59^{9} + \left(5 a^{2} + 29 a + 26\right)\cdot 59^{10} + \left(17 a^{2} + 45 a + 25\right)\cdot 59^{11} + \left(8 a^{2} + 33 a\right)\cdot 59^{12} + \left(27 a^{2} + 46 a + 18\right)\cdot 59^{13} + \left(35 a^{2} + 50 a + 48\right)\cdot 59^{14} +O\left(59^{ 15 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 24 a^{2} + 12 a + 17 + \left(14 a^{2} + 47 a + 22\right)\cdot 59 + \left(52 a^{2} + a + 22\right)\cdot 59^{2} + \left(51 a^{2} + 42 a + 11\right)\cdot 59^{3} + \left(56 a^{2} + 34 a + 6\right)\cdot 59^{4} + \left(50 a + 58\right)\cdot 59^{5} + \left(35 a^{2} + 22 a + 7\right)\cdot 59^{6} + \left(43 a^{2} + 7 a + 53\right)\cdot 59^{7} + \left(44 a^{2} + 17 a + 4\right)\cdot 59^{8} + \left(20 a + 28\right)\cdot 59^{9} + \left(32 a^{2} + 29 a + 36\right)\cdot 59^{10} + \left(15 a^{2} + 54 a\right)\cdot 59^{11} + \left(55 a^{2} + 48 a + 39\right)\cdot 59^{12} + \left(34 a^{2} + 50 a + 4\right)\cdot 59^{13} + \left(25 a^{2} + 34 a + 35\right)\cdot 59^{14} +O\left(59^{ 15 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 12 + 34\cdot 59 + 44\cdot 59^{2} + 11\cdot 59^{3} + 25\cdot 59^{4} + 34\cdot 59^{5} + 33\cdot 59^{6} + 47\cdot 59^{7} + 26\cdot 59^{8} + 14\cdot 59^{9} + 19\cdot 59^{10} + 11\cdot 59^{11} + 16\cdot 59^{12} + 8\cdot 59^{13} + 57\cdot 59^{14} +O\left(59^{ 15 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,2,4)(3,5,9)(6,7,8)$ |
| $(1,8,9)(2,6,3)(4,7,5)$ |
| $(1,2,4)(3,9,5)$ |
| $(1,9)(2,5)(3,4)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,9)(2,5)(3,4)(6,7)$ |
$0$ |
| $2$ |
$3$ |
$(1,2,4)(3,5,9)(6,7,8)$ |
$-3$ |
| $3$ |
$3$ |
$(3,9,5)(6,7,8)$ |
$0$ |
| $3$ |
$3$ |
$(3,5,9)(6,8,7)$ |
$0$ |
| $6$ |
$3$ |
$(1,8,9)(2,6,3)(4,7,5)$ |
$0$ |
| $6$ |
$3$ |
$(1,8,5)(2,6,9)(3,4,7)$ |
$0$ |
| $6$ |
$3$ |
$(1,5,8)(2,9,6)(3,7,4)$ |
$0$ |
| $9$ |
$6$ |
$(1,2)(3,7,9,8,5,6)$ |
$0$ |
| $9$ |
$6$ |
$(1,2)(3,6,5,8,9,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
