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:
| $r_{ 1 }$ |
$=$ |
$ 11 a + 32 + \left(35 a + 27\right)\cdot 59 + \left(30 a + 27\right)\cdot 59^{2} + \left(17 a + 36\right)\cdot 59^{3} + \left(15 a + 55\right)\cdot 59^{4} + \left(8 a + 30\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 48 a + 43 + \left(23 a + 51\right)\cdot 59 + \left(28 a + 22\right)\cdot 59^{2} + \left(41 a + 23\right)\cdot 59^{3} + \left(43 a + 53\right)\cdot 59^{4} + \left(50 a + 23\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 45 a + 11 + \left(25 a + 30\right)\cdot 59 + \left(21 a + 17\right)\cdot 59^{2} + \left(a + 3\right)\cdot 59^{3} + \left(35 a + 45\right)\cdot 59^{4} + \left(5 a + 51\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 50 + 9\cdot 59 + 52\cdot 59^{2} + 17\cdot 59^{3} + 59^{4} + 47\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 a + 58 + \left(3 a + 16\right)\cdot 59 + \left(12 a + 26\right)\cdot 59^{2} + \left(13 a + 57\right)\cdot 59^{3} + \left(10 a + 25\right)\cdot 59^{4} + \left(16 a + 23\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 + 26\cdot 59 + 41\cdot 59^{2} + 55\cdot 59^{3} + 11\cdot 59^{4} + 7\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 42 a + 16 + \left(55 a + 3\right)\cdot 59 + \left(46 a + 35\right)\cdot 59^{2} + \left(45 a + 58\right)\cdot 59^{3} + \left(48 a + 22\right)\cdot 59^{4} + \left(42 a + 29\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 14 a + 56 + \left(33 a + 10\right)\cdot 59 + \left(37 a + 13\right)\cdot 59^{2} + \left(57 a + 42\right)\cdot 59^{3} + \left(23 a + 19\right)\cdot 59^{4} + \left(53 a + 22\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,2)(5,6,7)$ |
| $(1,5)(2,7)(3,8)(4,6)$ |
| $(1,5)(2,6)(4,7)$ |
| $(1,8,5,3)(2,4,7,6)$ |
| $(1,2,5,7)(3,4,8,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,5)(2,7)(3,8)(4,6)$ |
$-4$ |
| $12$ |
$2$ |
$(1,5)(2,6)(4,7)$ |
$0$ |
| $8$ |
$3$ |
$(2,6,3)(4,8,7)$ |
$1$ |
| $6$ |
$4$ |
$(1,2,5,7)(3,4,8,6)$ |
$0$ |
| $8$ |
$6$ |
$(1,6,2,5,4,7)(3,8)$ |
$-1$ |
| $6$ |
$8$ |
$(1,6,3,7,5,4,8,2)$ |
$0$ |
| $6$ |
$8$ |
$(1,4,3,2,5,6,8,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
