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