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