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