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:
| $r_{ 1 }$ |
$=$ |
$ 49 a + 5 + \left(50 a + 58\right)\cdot 59 + \left(56 a + 55\right)\cdot 59^{2} + \left(6 a + 24\right)\cdot 59^{3} + \left(15 a + 25\right)\cdot 59^{4} + \left(6 a + 4\right)\cdot 59^{5} + \left(58 a + 33\right)\cdot 59^{6} + \left(42 a + 7\right)\cdot 59^{7} + \left(33 a + 34\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 a + 18 + \left(14 a + 21\right)\cdot 59 + \left(52 a + 51\right)\cdot 59^{2} + \left(7 a + 6\right)\cdot 59^{3} + \left(50 a + 29\right)\cdot 59^{4} + \left(7 a + 41\right)\cdot 59^{5} + \left(52 a + 56\right)\cdot 59^{6} + \left(6 a + 29\right)\cdot 59^{7} + \left(14 a + 36\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 a + 19 + \left(14 a + 45\right)\cdot 59 + \left(52 a + 28\right)\cdot 59^{2} + \left(7 a + 37\right)\cdot 59^{3} + \left(50 a + 46\right)\cdot 59^{4} + 7 a\cdot 59^{5} + \left(52 a + 17\right)\cdot 59^{6} + \left(6 a + 15\right)\cdot 59^{7} + \left(14 a + 15\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 + 19\cdot 59 + 49\cdot 59^{2} + 22\cdot 59^{3} + 30\cdot 59^{4} + 32\cdot 59^{5} + 10\cdot 59^{6} + 52\cdot 59^{7} + 23\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a + 54 + 8 a\cdot 59 + \left(2 a + 3\right)\cdot 59^{2} + \left(52 a + 34\right)\cdot 59^{3} + \left(43 a + 33\right)\cdot 59^{4} + \left(52 a + 54\right)\cdot 59^{5} + 25\cdot 59^{6} + \left(16 a + 51\right)\cdot 59^{7} + \left(25 a + 24\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 a + 41 + \left(44 a + 37\right)\cdot 59 + \left(6 a + 7\right)\cdot 59^{2} + \left(51 a + 52\right)\cdot 59^{3} + \left(8 a + 29\right)\cdot 59^{4} + \left(51 a + 17\right)\cdot 59^{5} + \left(6 a + 2\right)\cdot 59^{6} + \left(52 a + 29\right)\cdot 59^{7} + \left(44 a + 22\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 37 a + 40 + \left(44 a + 13\right)\cdot 59 + \left(6 a + 30\right)\cdot 59^{2} + \left(51 a + 21\right)\cdot 59^{3} + \left(8 a + 12\right)\cdot 59^{4} + \left(51 a + 58\right)\cdot 59^{5} + \left(6 a + 41\right)\cdot 59^{6} + \left(52 a + 43\right)\cdot 59^{7} + \left(44 a + 43\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 36 + 39\cdot 59 + 9\cdot 59^{2} + 36\cdot 59^{3} + 28\cdot 59^{4} + 26\cdot 59^{5} + 48\cdot 59^{6} + 6\cdot 59^{7} + 35\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,5,7)(2,8,6,4)$ |
| $(1,5)(2,4)(6,8)$ |
| $(1,8,5,4)(2,7,6,3)$ |
| $(1,2,8)(4,5,6)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-2$ |
| $12$ | $2$ | $(1,5)(2,4)(6,8)$ | $0$ |
| $8$ | $3$ | $(1,2,8)(4,5,6)$ | $-1$ |
| $6$ | $4$ | $(1,3,5,7)(2,8,6,4)$ | $0$ |
| $8$ | $6$ | $(1,8,3,5,4,7)(2,6)$ | $1$ |
| $6$ | $8$ | $(1,4,7,6,5,8,3,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,8,7,2,5,4,3,6)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
