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