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