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