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