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