Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22 a + 23 + \left(18 a + 27\right)\cdot 29 + \left(16 a + 25\right)\cdot 29^{2} + \left(7 a + 25\right)\cdot 29^{3} + \left(16 a + 22\right)\cdot 29^{4} + \left(3 a + 14\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 a + 15 + \left(16 a + 23\right)\cdot 29 + \left(24 a + 25\right)\cdot 29^{2} + \left(15 a + 1\right)\cdot 29^{3} + \left(26 a + 13\right)\cdot 29^{4} + \left(23 a + 14\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 17 + \left(10 a + 12\right)\cdot 29 + \left(12 a + 3\right)\cdot 29^{2} + \left(21 a + 18\right)\cdot 29^{3} + \left(12 a + 9\right)\cdot 29^{4} + \left(25 a + 16\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 + 9\cdot 29 + 20\cdot 29^{2} + 7\cdot 29^{3} + 24\cdot 29^{4} + 16\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 a + 7 + \left(12 a + 8\right)\cdot 29 + \left(4 a + 16\right)\cdot 29^{2} + \left(13 a + 27\right)\cdot 29^{3} + \left(2 a + 13\right)\cdot 29^{4} + \left(5 a + 20\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 5 + 5\cdot 29 + 24\cdot 29^{2} + 5\cdot 29^{3} + 3\cdot 29^{4} + 4\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(4,6)$ |
| $(1,6,5)(2,3,4)$ |
| $(2,5)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $3$ |
$2$ |
$(1,3)(4,6)$ |
$-1$ |
| $4$ |
$3$ |
$(1,6,5)(2,3,4)$ |
$0$ |
| $4$ |
$3$ |
$(1,5,6)(2,4,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
