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:
| $r_{ 1 }$ |
$=$ |
$ 18 a + 16 + \left(3 a + 8\right)\cdot 59 + \left(5 a + 49\right)\cdot 59^{2} + \left(57 a + 2\right)\cdot 59^{3} + \left(6 a + 2\right)\cdot 59^{4} + \left(36 a + 44\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 48 a + 30 + \left(30 a + 39\right)\cdot 59 + \left(a + 39\right)\cdot 59^{2} + \left(5 a + 44\right)\cdot 59^{3} + \left(13 a + 11\right)\cdot 59^{4} + \left(43 a + 56\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 a + 44 + \left(19 a + 44\right)\cdot 59 + \left(25 a + 28\right)\cdot 59^{2} + \left(51 a + 41\right)\cdot 59^{3} + \left(42 a + 32\right)\cdot 59^{4} + \left(57 a + 33\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 44 + 41\cdot 59 + 50\cdot 59^{2} + 7\cdot 59^{3} + 32\cdot 59^{4} + 25\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 a + 19 + \left(28 a + 22\right)\cdot 59 + \left(57 a + 10\right)\cdot 59^{2} + \left(53 a + 48\right)\cdot 59^{3} + \left(45 a + 19\right)\cdot 59^{4} + \left(15 a + 27\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 a + 16 + \left(39 a + 33\right)\cdot 59 + \left(33 a + 34\right)\cdot 59^{2} + \left(7 a + 8\right)\cdot 59^{3} + \left(16 a + 24\right)\cdot 59^{4} + \left(a + 48\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 36 + 52\cdot 59 + 30\cdot 59^{2} + 27\cdot 59^{3} + 43\cdot 59^{4} + 45\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 41 a + 34 + \left(55 a + 52\right)\cdot 59 + \left(53 a + 50\right)\cdot 59^{2} + \left(a + 54\right)\cdot 59^{3} + \left(52 a + 10\right)\cdot 59^{4} + \left(22 a + 14\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,6,2)(3,7,8,4)$ |
| $(1,8,6,3)(2,4,5,7)$ |
| $(1,3)(4,7)(6,8)$ |
| $(1,6)(2,5)(3,8)(4,7)$ |
| $(1,8,4)(3,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $-2$ |
| $12$ | $2$ | $(1,3)(4,7)(6,8)$ | $0$ |
| $8$ | $3$ | $(1,3,2)(5,6,8)$ | $-1$ |
| $6$ | $4$ | $(1,5,6,2)(3,7,8,4)$ | $0$ |
| $8$ | $6$ | $(1,5,3,6,2,8)(4,7)$ | $1$ |
| $6$ | $8$ | $(1,5,8,7,6,2,3,4)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,2,8,4,6,5,3,7)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
