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