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:
| $r_{ 1 }$ |
$=$ |
$ 42 + 23\cdot 59 + 26\cdot 59^{2} + 20\cdot 59^{3} + 32\cdot 59^{4} + 57\cdot 59^{5} + 24\cdot 59^{6} + 11\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 a + 25 + \left(10 a + 40\right)\cdot 59 + \left(16 a + 35\right)\cdot 59^{2} + \left(7 a + 39\right)\cdot 59^{3} + \left(21 a + 29\right)\cdot 59^{4} + \left(27 a + 36\right)\cdot 59^{5} + 3\cdot 59^{6} + \left(27 a + 15\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 a + \left(48 a + 17\right)\cdot 59 + \left(42 a + 41\right)\cdot 59^{2} + \left(51 a + 30\right)\cdot 59^{3} + \left(37 a + 43\right)\cdot 59^{4} + \left(31 a + 42\right)\cdot 59^{5} + \left(58 a + 35\right)\cdot 59^{6} + \left(31 a + 41\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a + 58 + \left(41 a + 9\right)\cdot 59 + \left(37 a + 31\right)\cdot 59^{2} + \left(37 a + 29\right)\cdot 59^{3} + \left(27 a + 34\right)\cdot 59^{4} + \left(36 a + 54\right)\cdot 59^{5} + \left(46 a + 53\right)\cdot 59^{6} + \left(3 a + 50\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 + 35\cdot 59 + 32\cdot 59^{2} + 38\cdot 59^{3} + 26\cdot 59^{4} + 59^{5} + 34\cdot 59^{6} + 47\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 a + 34 + \left(48 a + 18\right)\cdot 59 + \left(42 a + 23\right)\cdot 59^{2} + \left(51 a + 19\right)\cdot 59^{3} + \left(37 a + 29\right)\cdot 59^{4} + \left(31 a + 22\right)\cdot 59^{5} + \left(58 a + 55\right)\cdot 59^{6} + \left(31 a + 43\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 34 a + \left(10 a + 42\right)\cdot 59 + \left(16 a + 17\right)\cdot 59^{2} + \left(7 a + 28\right)\cdot 59^{3} + \left(21 a + 15\right)\cdot 59^{4} + \left(27 a + 16\right)\cdot 59^{5} + 23\cdot 59^{6} + \left(27 a + 17\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 57 a + 1 + \left(17 a + 49\right)\cdot 59 + \left(21 a + 27\right)\cdot 59^{2} + \left(21 a + 29\right)\cdot 59^{3} + \left(31 a + 24\right)\cdot 59^{4} + \left(22 a + 4\right)\cdot 59^{5} + \left(12 a + 5\right)\cdot 59^{6} + \left(55 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 |
| $(1,2,5,6)(3,4,7,8)$ |
| $(1,5)(3,8)(4,7)$ |
| $(1,3,4)(5,7,8)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
| $(1,4,5,8)(2,3,6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$-4$ |
| $12$ |
$2$ |
$(1,5)(3,8)(4,7)$ |
$0$ |
| $8$ |
$3$ |
$(2,4,7)(3,6,8)$ |
$1$ |
| $6$ |
$4$ |
$(1,4,5,8)(2,3,6,7)$ |
$0$ |
| $8$ |
$6$ |
$(1,4,2,5,8,6)(3,7)$ |
$-1$ |
| $6$ |
$8$ |
$(1,7,2,8,5,3,6,4)$ |
$0$ |
| $6$ |
$8$ |
$(1,3,2,4,5,7,6,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
