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:
| $r_{ 1 }$ |
$=$ |
$ 40 a + 48 + \left(47 a + 50\right)\cdot 59 + \left(38 a + 34\right)\cdot 59^{2} + \left(54 a + 25\right)\cdot 59^{3} + \left(20 a + 21\right)\cdot 59^{4} + \left(39 a + 44\right)\cdot 59^{5} + \left(35 a + 27\right)\cdot 59^{6} + \left(25 a + 34\right)\cdot 59^{7} + \left(7 a + 31\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 a + 29 + \left(11 a + 58\right)\cdot 59 + \left(20 a + 25\right)\cdot 59^{2} + \left(4 a + 41\right)\cdot 59^{3} + \left(38 a + 46\right)\cdot 59^{4} + \left(19 a + 3\right)\cdot 59^{5} + \left(23 a + 24\right)\cdot 59^{6} + \left(33 a + 24\right)\cdot 59^{7} + \left(51 a + 13\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 + 28\cdot 59 + 11\cdot 59^{2} + 23\cdot 59^{3} + 34\cdot 59^{4} + 11\cdot 59^{5} + 27\cdot 59^{6} + 7\cdot 59^{7} + 43\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 a + 11 + \left(11 a + 8\right)\cdot 59 + \left(20 a + 24\right)\cdot 59^{2} + \left(4 a + 33\right)\cdot 59^{3} + \left(38 a + 37\right)\cdot 59^{4} + \left(19 a + 14\right)\cdot 59^{5} + \left(23 a + 31\right)\cdot 59^{6} + \left(33 a + 24\right)\cdot 59^{7} + \left(51 a + 27\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 a + 30 + 47 a\cdot 59 + \left(38 a + 33\right)\cdot 59^{2} + \left(54 a + 17\right)\cdot 59^{3} + \left(20 a + 12\right)\cdot 59^{4} + \left(39 a + 55\right)\cdot 59^{5} + \left(35 a + 34\right)\cdot 59^{6} + \left(25 a + 34\right)\cdot 59^{7} + \left(7 a + 45\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 48 + 30\cdot 59 + 47\cdot 59^{2} + 35\cdot 59^{3} + 24\cdot 59^{4} + 47\cdot 59^{5} + 31\cdot 59^{6} + 51\cdot 59^{7} + 15\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,5)$ |
| $(2,5)$ |
| $(1,3,2)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-3$ |
| $3$ |
$2$ |
$(1,4)(2,5)$ |
$-1$ |
| $3$ |
$2$ |
$(2,5)$ |
$1$ |
| $6$ |
$2$ |
$(1,2)(4,5)$ |
$1$ |
| $6$ |
$2$ |
$(1,4)(2,3)(5,6)$ |
$-1$ |
| $8$ |
$3$ |
$(1,3,2)(4,6,5)$ |
$0$ |
| $6$ |
$4$ |
$(1,5,4,2)$ |
$1$ |
| $6$ |
$4$ |
$(1,4)(2,6,5,3)$ |
$-1$ |
| $8$ |
$6$ |
$(1,3,2,4,6,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
