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:
| $r_{ 1 }$ |
$=$ |
$ 15 a + 17 + \left(a + 20\right)\cdot 29 + \left(4 a + 3\right)\cdot 29^{2} + \left(11 a + 27\right)\cdot 29^{3} + \left(10 a + 5\right)\cdot 29^{4} + \left(4 a + 4\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 26\cdot 29 + 20\cdot 29^{2} + 18\cdot 29^{3} + 23\cdot 29^{4} + 18\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 a + \left(15 a + 2\right)\cdot 29 + \left(11 a + 26\right)\cdot 29^{2} + \left(23 a + 22\right)\cdot 29^{3} + \left(6 a + 13\right)\cdot 29^{4} + \left(8 a + 3\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 14 a + 5 + \left(27 a + 13\right)\cdot 29 + \left(24 a + 22\right)\cdot 29^{2} + \left(17 a + 20\right)\cdot 29^{3} + \left(18 a + 17\right)\cdot 29^{4} + \left(24 a + 15\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 a + 9 + \left(13 a + 27\right)\cdot 29 + \left(17 a + 9\right)\cdot 29^{2} + \left(5 a + 12\right)\cdot 29^{3} + \left(22 a + 24\right)\cdot 29^{4} + \left(20 a + 8\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 a + 20 a\cdot 29 + \left(24 a + 13\right)\cdot 29^{2} + \left(25 a + 28\right)\cdot 29^{3} + \left(4 a + 27\right)\cdot 29^{4} + \left(23 a + 22\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 9 a + 13 + \left(8 a + 25\right)\cdot 29 + \left(4 a + 28\right)\cdot 29^{2} + \left(3 a + 16\right)\cdot 29^{3} + \left(24 a + 26\right)\cdot 29^{4} + \left(5 a + 17\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 11 + 29 + 20\cdot 29^{2} + 26\cdot 29^{3} + 4\cdot 29^{4} + 24\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,5,3)(2,7,8,6)$ |
| $(1,5)(2,8)(3,4)(6,7)$ |
| $(1,2)(3,4)(5,8)$ |
| $(1,4,8)(2,5,3)$ |
| $(1,6,5,7)(2,4,8,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,8)(3,4)(6,7)$ | $-4$ |
| $12$ | $2$ | $(1,2)(3,4)(5,8)$ | $0$ |
| $8$ | $3$ | $(1,3,7)(4,6,5)$ | $1$ |
| $6$ | $4$ | $(1,6,5,7)(2,4,8,3)$ | $0$ |
| $8$ | $6$ | $(1,6,3,5,7,4)(2,8)$ | $-1$ |
| $6$ | $8$ | $(1,4,2,6,5,3,8,7)$ | $0$ |
| $6$ | $8$ | $(1,3,2,7,5,4,8,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
