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