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:
| $r_{ 1 }$ |
$=$ |
$ 32 + 15\cdot 59 + 54\cdot 59^{2} + 19\cdot 59^{3} + 16\cdot 59^{4} + 12\cdot 59^{5} + 33\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 50 a + 34 + \left(40 a + 4\right)\cdot 59 + \left(50 a + 54\right)\cdot 59^{2} + \left(28 a + 10\right)\cdot 59^{3} + \left(34 a + 56\right)\cdot 59^{4} + \left(58 a + 46\right)\cdot 59^{5} + \left(44 a + 6\right)\cdot 59^{6} + \left(36 a + 4\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a + 14 + \left(21 a + 42\right)\cdot 59 + \left(23 a + 1\right)\cdot 59^{2} + \left(10 a + 11\right)\cdot 59^{3} + \left(15 a + 42\right)\cdot 59^{4} + \left(16 a + 1\right)\cdot 59^{5} + \left(14 a + 20\right)\cdot 59^{6} + \left(44 a + 37\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 a + 36 + \left(21 a + 4\right)\cdot 59 + \left(23 a + 55\right)\cdot 59^{2} + \left(10 a + 1\right)\cdot 59^{3} + \left(15 a + 12\right)\cdot 59^{4} + \left(16 a + 56\right)\cdot 59^{5} + \left(14 a + 40\right)\cdot 59^{6} + \left(44 a + 50\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 + 43\cdot 59 + 4\cdot 59^{2} + 39\cdot 59^{3} + 42\cdot 59^{4} + 46\cdot 59^{5} + 58\cdot 59^{6} + 25\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 a + 25 + \left(18 a + 54\right)\cdot 59 + \left(8 a + 4\right)\cdot 59^{2} + \left(30 a + 48\right)\cdot 59^{3} + \left(24 a + 2\right)\cdot 59^{4} + 12\cdot 59^{5} + \left(14 a + 52\right)\cdot 59^{6} + \left(22 a + 54\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 50 a + 45 + \left(37 a + 16\right)\cdot 59 + \left(35 a + 57\right)\cdot 59^{2} + \left(48 a + 47\right)\cdot 59^{3} + \left(43 a + 16\right)\cdot 59^{4} + \left(42 a + 57\right)\cdot 59^{5} + \left(44 a + 38\right)\cdot 59^{6} + \left(14 a + 21\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 50 a + 23 + \left(37 a + 54\right)\cdot 59 + \left(35 a + 3\right)\cdot 59^{2} + \left(48 a + 57\right)\cdot 59^{3} + \left(43 a + 46\right)\cdot 59^{4} + \left(42 a + 2\right)\cdot 59^{5} + \left(44 a + 18\right)\cdot 59^{6} + \left(14 a + 8\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,6)(2,5,8)$ |
| $(1,5)(2,4)(6,8)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
| $(1,7,5,3)(2,4,6,8)$ |
| $(1,4,5,8)(2,3,6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$-4$ |
| $12$ |
$2$ |
$(1,5)(2,4)(6,8)$ |
$0$ |
| $8$ |
$3$ |
$(1,8,3)(4,7,5)$ |
$1$ |
| $6$ |
$4$ |
$(1,7,5,3)(2,4,6,8)$ |
$0$ |
| $8$ |
$6$ |
$(1,7,8,5,3,4)(2,6)$ |
$-1$ |
| $6$ |
$8$ |
$(1,8,7,2,5,4,3,6)$ |
$0$ |
| $6$ |
$8$ |
$(1,4,7,6,5,8,3,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
