Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 a + 27 + \left(19 a + 14\right)\cdot 29 + \left(23 a + 15\right)\cdot 29^{2} + \left(16 a + 12\right)\cdot 29^{3} + \left(8 a + 14\right)\cdot 29^{4} + \left(23 a + 8\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 a + 5 + \left(13 a + 14\right)\cdot 29 + \left(26 a + 25\right)\cdot 29^{2} + \left(4 a + 11\right)\cdot 29^{3} + \left(6 a + 14\right)\cdot 29^{4} + \left(7 a + 28\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 a + 17 + \left(5 a + 12\right)\cdot 29 + \left(23 a + 14\right)\cdot 29^{2} + 17\cdot 29^{3} + \left(7 a + 27\right)\cdot 29^{4} + \left(5 a + 27\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 + 2\cdot 29 + 25\cdot 29^{2} + 5\cdot 29^{3} + 4\cdot 29^{4} + 17\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 + 21\cdot 29 + 28\cdot 29^{2} + 15\cdot 29^{3} + 28\cdot 29^{4} + 14\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 5 a + 21 + \left(23 a + 17\right)\cdot 29 + \left(5 a + 8\right)\cdot 29^{2} + \left(28 a + 27\right)\cdot 29^{3} + \left(21 a + 3\right)\cdot 29^{4} + \left(23 a + 18\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 19 a + 19 + \left(9 a + 14\right)\cdot 29 + \left(5 a + 27\right)\cdot 29^{2} + \left(12 a + 14\right)\cdot 29^{3} + \left(20 a + 11\right)\cdot 29^{4} + 5 a\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 25 a + 25 + \left(15 a + 17\right)\cdot 29 + \left(2 a + 28\right)\cdot 29^{2} + \left(24 a + 9\right)\cdot 29^{3} + \left(22 a + 11\right)\cdot 29^{4} + 21 a\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(2,7)(4,8)$ |
| $(1,7,4)(2,5,8)$ |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,6,7,3)$ |
| $(1,2,8,7)(3,5,6,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-4$ |
| $12$ |
$2$ |
$(1,5)(2,7)(4,8)$ |
$0$ |
| $8$ |
$3$ |
$(1,7,4)(2,5,8)$ |
$1$ |
| $6$ |
$4$ |
$(1,4,8,5)(2,6,7,3)$ |
$0$ |
| $8$ |
$6$ |
$(1,2,4,8,7,5)(3,6)$ |
$-1$ |
| $6$ |
$8$ |
$(1,6,4,7,8,3,5,2)$ |
$0$ |
| $6$ |
$8$ |
$(1,3,4,2,8,6,5,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
