Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 55 a + 24 + \left(19 a + 19\right)\cdot 59 + \left(34 a + 43\right)\cdot 59^{2} + \left(28 a + 54\right)\cdot 59^{3} + \left(40 a + 46\right)\cdot 59^{4} + \left(56 a + 38\right)\cdot 59^{5} + \left(12 a + 22\right)\cdot 59^{6} + \left(17 a + 24\right)\cdot 59^{7} + \left(51 a + 55\right)\cdot 59^{8} + \left(6 a + 44\right)\cdot 59^{9} +O\left(59^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 17\cdot 59 + 33\cdot 59^{2} + 6\cdot 59^{3} + 21\cdot 59^{4} + 4\cdot 59^{5} + 39\cdot 59^{6} + 43\cdot 59^{7} + 49\cdot 59^{8} + 10\cdot 59^{9} +O\left(59^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 55 a + 40 + \left(19 a + 15\right)\cdot 59 + \left(34 a + 1\right)\cdot 59^{2} + \left(28 a + 10\right)\cdot 59^{3} + 40 a\cdot 59^{4} + \left(56 a + 4\right)\cdot 59^{5} + \left(12 a + 21\right)\cdot 59^{6} + \left(17 a + 30\right)\cdot 59^{7} + \left(51 a + 28\right)\cdot 59^{8} + \left(6 a + 58\right)\cdot 59^{9} +O\left(59^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a + 20 + \left(39 a + 43\right)\cdot 59 + \left(24 a + 57\right)\cdot 59^{2} + \left(30 a + 48\right)\cdot 59^{3} + \left(18 a + 58\right)\cdot 59^{4} + \left(2 a + 54\right)\cdot 59^{5} + \left(46 a + 37\right)\cdot 59^{6} + \left(41 a + 28\right)\cdot 59^{7} + \left(7 a + 30\right)\cdot 59^{8} + 52 a\cdot 59^{9} +O\left(59^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 a + 36 + \left(39 a + 39\right)\cdot 59 + \left(24 a + 15\right)\cdot 59^{2} + \left(30 a + 4\right)\cdot 59^{3} + \left(18 a + 12\right)\cdot 59^{4} + \left(2 a + 20\right)\cdot 59^{5} + \left(46 a + 36\right)\cdot 59^{6} + \left(41 a + 34\right)\cdot 59^{7} + \left(7 a + 3\right)\cdot 59^{8} + \left(52 a + 14\right)\cdot 59^{9} +O\left(59^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 52 + 41\cdot 59 + 25\cdot 59^{2} + 52\cdot 59^{3} + 37\cdot 59^{4} + 54\cdot 59^{5} + 19\cdot 59^{6} + 15\cdot 59^{7} + 9\cdot 59^{8} + 48\cdot 59^{9} +O\left(59^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(5,6)$ |
| $(1,5)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,4)$ |
$-3$ |
| $3$ |
$2$ |
$(3,4)$ |
$1$ |
| $3$ |
$2$ |
$(1,5)(3,4)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)(5,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$1$ |
| $8$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,3,5,4)$ |
$-1$ |
| $6$ |
$4$ |
$(1,3,5,4)(2,6)$ |
$1$ |
| $8$ |
$6$ |
$(1,2,3,5,6,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
