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:
| $r_{ 1 }$ |
$=$ |
$ 52 a + 38 + \left(7 a + 56\right)\cdot 59 + \left(a + 38\right)\cdot 59^{2} + \left(44 a + 53\right)\cdot 59^{3} + \left(27 a + 48\right)\cdot 59^{4} + \left(54 a + 26\right)\cdot 59^{5} + 15\cdot 59^{6} + \left(9 a + 3\right)\cdot 59^{7} + \left(22 a + 24\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 52 a + 28 + \left(7 a + 46\right)\cdot 59 + \left(a + 26\right)\cdot 59^{2} + \left(44 a + 21\right)\cdot 59^{3} + \left(27 a + 26\right)\cdot 59^{4} + \left(54 a + 5\right)\cdot 59^{5} + 38\cdot 59^{6} + \left(9 a + 47\right)\cdot 59^{7} + \left(22 a + 21\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 a + 56 + \left(36 a + 43\right)\cdot 59 + \left(8 a + 13\right)\cdot 59^{2} + \left(23 a + 22\right)\cdot 59^{3} + \left(58 a + 41\right)\cdot 59^{4} + \left(4 a + 26\right)\cdot 59^{5} + \left(6 a + 58\right)\cdot 59^{6} + \left(17 a + 23\right)\cdot 59^{7} + \left(27 a + 24\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 + 47\cdot 59 + 17\cdot 59^{2} + 46\cdot 59^{3} + 21\cdot 59^{4} + 22\cdot 59^{5} + 22\cdot 59^{6} + 21\cdot 59^{7} + 30\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 a + 21 + \left(51 a + 2\right)\cdot 59 + \left(57 a + 20\right)\cdot 59^{2} + \left(14 a + 5\right)\cdot 59^{3} + \left(31 a + 10\right)\cdot 59^{4} + \left(4 a + 32\right)\cdot 59^{5} + \left(58 a + 43\right)\cdot 59^{6} + \left(49 a + 55\right)\cdot 59^{7} + \left(36 a + 34\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 a + 31 + \left(51 a + 12\right)\cdot 59 + \left(57 a + 32\right)\cdot 59^{2} + \left(14 a + 37\right)\cdot 59^{3} + \left(31 a + 32\right)\cdot 59^{4} + \left(4 a + 53\right)\cdot 59^{5} + \left(58 a + 20\right)\cdot 59^{6} + \left(49 a + 11\right)\cdot 59^{7} + \left(36 a + 37\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 53 a + 3 + \left(22 a + 15\right)\cdot 59 + \left(50 a + 45\right)\cdot 59^{2} + \left(35 a + 36\right)\cdot 59^{3} + 17\cdot 59^{4} + \left(54 a + 32\right)\cdot 59^{5} + 52 a\cdot 59^{6} + \left(41 a + 35\right)\cdot 59^{7} + \left(31 a + 34\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 52 + 11\cdot 59 + 41\cdot 59^{2} + 12\cdot 59^{3} + 37\cdot 59^{4} + 36\cdot 59^{5} + 36\cdot 59^{6} + 37\cdot 59^{7} + 28\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(4,8)(5,6)$ |
| $(1,6,4)(2,8,5)$ |
| $(1,8,5,4)(2,3,6,7)$ |
| $(1,7,5,3)(2,8,6,4)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-2$ |
| $12$ | $2$ | $(1,2)(4,8)(5,6)$ | $0$ |
| $8$ | $3$ | $(2,4,7)(3,6,8)$ | $-1$ |
| $6$ | $4$ | $(1,8,5,4)(2,3,6,7)$ | $0$ |
| $8$ | $6$ | $(1,5)(2,3,4,6,7,8)$ | $1$ |
| $6$ | $8$ | $(1,3,6,4,5,7,2,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,7,6,8,5,3,2,4)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
