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