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