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