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:
| $r_{ 1 }$ |
$=$ |
$ 11 a + 11 + \left(4 a + 9\right)\cdot 29 + \left(18 a + 22\right)\cdot 29^{2} + \left(25 a + 10\right)\cdot 29^{3} + \left(20 a + 11\right)\cdot 29^{4} + \left(17 a + 18\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 a + \left(6 a + 22\right)\cdot 29 + 26 a\cdot 29^{2} + \left(19 a + 27\right)\cdot 29^{3} + \left(a + 4\right)\cdot 29^{4} + 5 a\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a + 8 + \left(24 a + 20\right)\cdot 29 + \left(10 a + 21\right)\cdot 29^{2} + \left(3 a + 4\right)\cdot 29^{3} + \left(8 a + 3\right)\cdot 29^{4} + \left(11 a + 28\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 a + 7 + \left(22 a + 12\right)\cdot 29 + \left(2 a + 9\right)\cdot 29^{2} + \left(9 a + 13\right)\cdot 29^{3} + \left(27 a + 22\right)\cdot 29^{4} + \left(23 a + 23\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 + 10\cdot 29 + 23\cdot 29^{2} + 22\cdot 29^{3} + 6\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 + 8\cdot 29 + 20\cdot 29^{2} + 20\cdot 29^{3} + 19\cdot 29^{4} + 17\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 12 a + 8 + \left(20 a + 15\right)\cdot 29 + \left(9 a + 9\right)\cdot 29^{2} + \left(13 a + 23\right)\cdot 29^{3} + \left(9 a + 9\right)\cdot 29^{4} + \left(24 a + 27\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 17 a + 10 + \left(8 a + 18\right)\cdot 29 + \left(19 a + 8\right)\cdot 29^{2} + \left(15 a + 22\right)\cdot 29^{3} + \left(19 a + 14\right)\cdot 29^{4} + \left(4 a + 23\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,8)(4,7)$ |
| $(1,4,8)(2,3,7)$ |
| $(1,7)(2,4)(3,8)(5,6)$ |
| $(1,3,7,8)(2,5,4,6)$ |
| $(1,2,7,4)(3,6,8,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,4)(3,8)(5,6)$ | $-2$ |
| $12$ | $2$ | $(1,2)(3,8)(4,7)$ | $0$ |
| $8$ | $3$ | $(1,4,8)(2,3,7)$ | $-1$ |
| $6$ | $4$ | $(1,3,7,8)(2,5,4,6)$ | $0$ |
| $8$ | $6$ | $(1,6,2,7,5,4)(3,8)$ | $1$ |
| $6$ | $8$ | $(1,8,2,5,7,3,4,6)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,3,2,6,7,8,4,5)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
