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:
| $r_{ 1 }$ |
$=$ |
$ 19 a + 12 + \left(43 a + 49\right)\cdot 59 + \left(51 a + 14\right)\cdot 59^{2} + \left(9 a + 12\right)\cdot 59^{3} + \left(17 a + 33\right)\cdot 59^{4} + \left(42 a + 49\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 + 49\cdot 59 + 45\cdot 59^{2} + 33\cdot 59^{3} + 4\cdot 59^{4} + 50\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 51 a + 56 + \left(49 a + 8\right)\cdot 59 + \left(33 a + 35\right)\cdot 59^{2} + \left(33 a + 35\right)\cdot 59^{3} + \left(19 a + 40\right)\cdot 59^{4} + \left(22 a + 8\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 a + 28 + \left(27 a + 14\right)\cdot 59 + \left(46 a + 22\right)\cdot 59^{2} + \left(23 a + 20\right)\cdot 59^{3} + \left(57 a + 46\right)\cdot 59^{4} + \left(35 a + 4\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 + 11\cdot 59 + 34\cdot 59^{2} + 12\cdot 59^{3} + 23\cdot 59^{4} + 53\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 8 a + 48 + \left(9 a + 7\right)\cdot 59 + \left(25 a + 19\right)\cdot 59^{2} + \left(25 a + 35\right)\cdot 59^{3} + \left(39 a + 26\right)\cdot 59^{4} + \left(36 a + 11\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 39 a + 48 + \left(31 a + 21\right)\cdot 59 + \left(12 a + 41\right)\cdot 59^{2} + \left(35 a + 56\right)\cdot 59^{3} + \left(a + 20\right)\cdot 59^{4} + \left(23 a + 42\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 40 a + 31 + \left(15 a + 14\right)\cdot 59 + \left(7 a + 23\right)\cdot 59^{2} + \left(49 a + 29\right)\cdot 59^{3} + \left(41 a + 40\right)\cdot 59^{4} + \left(16 a + 15\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7)(2,5)(6,8)$ |
| $(1,6,5)(2,8,7)$ |
| $(1,8)(2,5)(3,4)(6,7)$ |
| $(1,2,8,5)(3,6,4,7)$ |
| $(1,4,8,3)(2,6,5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,5)(3,4)(6,7)$ | $-2$ |
| $12$ | $2$ | $(1,7)(2,5)(6,8)$ | $0$ |
| $8$ | $3$ | $(1,4,2)(3,5,8)$ | $-1$ |
| $6$ | $4$ | $(1,4,8,3)(2,6,5,7)$ | $0$ |
| $8$ | $6$ | $(1,5,4,8,2,3)(6,7)$ | $1$ |
| $6$ | $8$ | $(1,2,7,4,8,5,6,3)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,5,7,3,8,2,6,4)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
