Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 27 a + 49 + \left(55 a + 3\right)\cdot 59 + \left(38 a + 52\right)\cdot 59^{2} + \left(54 a + 18\right)\cdot 59^{3} + \left(44 a + 18\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 50 + 22\cdot 59 + 39\cdot 59^{2} + 53\cdot 59^{3} + 31\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 50 a + 21 + \left(a + 20\right)\cdot 59 + \left(14 a + 15\right)\cdot 59^{2} + \left(37 a + 54\right)\cdot 59^{3} + \left(35 a + 8\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 32 a + 17 + \left(3 a + 32\right)\cdot 59 + \left(20 a + 35\right)\cdot 59^{2} + \left(4 a + 34\right)\cdot 59^{3} + \left(14 a + 8\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 a + 1 + \left(16 a + 16\right)\cdot 59 + \left(13 a + 3\right)\cdot 59^{2} + \left(22 a + 30\right)\cdot 59^{3} + \left(54 a + 41\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 a + 12 + \left(57 a + 31\right)\cdot 59 + \left(44 a + 27\right)\cdot 59^{2} + \left(21 a + 18\right)\cdot 59^{3} + \left(23 a + 7\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 13 + 33\cdot 59 + 3\cdot 59^{2} + 46\cdot 59^{3} + 45\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 44 a + 16 + \left(42 a + 17\right)\cdot 59 + 45 a\cdot 59^{2} + \left(36 a + 39\right)\cdot 59^{3} + \left(4 a + 14\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3)(6,8,7)$ |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,6,7,3)$ |
| $(1,8)(2,6)(3,7)$ |
| $(1,6,8,3)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $12$ | $2$ | $(1,8)(2,6)(3,7)$ | $0$ |
| $8$ | $3$ | $(1,6,5)(3,4,8)$ | $-1$ |
| $6$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $8$ | $6$ | $(1,4,6,8,5,3)(2,7)$ | $1$ |
| $6$ | $8$ | $(1,3,4,2,8,6,5,7)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,6,4,7,8,3,5,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
