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:
| $r_{ 1 }$ |
$=$ |
$ 25 a + 23 + \left(27 a + 13\right)\cdot 29 + \left(9 a + 16\right)\cdot 29^{2} + \left(22 a + 19\right)\cdot 29^{3} + \left(16 a + 21\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 3\cdot 29 + 3\cdot 29^{2} + 4\cdot 29^{3} + 11\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a + 15 + \left(11 a + 14\right)\cdot 29 + \left(12 a + 23\right)\cdot 29^{2} + \left(28 a + 4\right)\cdot 29^{3} + \left(8 a + 6\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a + 3 + \left(a + 12\right)\cdot 29 + \left(19 a + 9\right)\cdot 29^{2} + \left(6 a + 5\right)\cdot 29^{3} + \left(12 a + 25\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 + 11\cdot 29 + 18\cdot 29^{2} + 5\cdot 29^{3} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 19 a + 7 + \left(17 a + 3\right)\cdot 29 + \left(16 a + 16\right)\cdot 29^{2} + 18\cdot 29^{3} + \left(20 a + 22\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,5)(4,6)$ |
| $(1,2)(3,5)$ |
| $(2,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,5)$ | $0$ |
| $3$ | $2$ | $(1,5)(2,3)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,4,2)(3,6,5)$ | $-1$ |
| $2$ | $6$ | $(1,6,2,3,4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
