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:
| $r_{ 1 }$ |
$=$ |
$ 6 a + 20 + \left(3 a + 7\right)\cdot 29 + \left(9 a + 24\right)\cdot 29^{2} + \left(19 a + 7\right)\cdot 29^{3} + \left(18 a + 16\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 + 12\cdot 29 + 20\cdot 29^{2} + 23\cdot 29^{3} + 8\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 a + 21 + \left(25 a + 17\right)\cdot 29 + \left(19 a + 8\right)\cdot 29^{2} + \left(9 a + 8\right)\cdot 29^{3} + \left(10 a + 3\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 a + 18 + \left(12 a + 28\right)\cdot 29 + 15 a\cdot 29^{2} + \left(5 a + 11\right)\cdot 29^{3} + \left(11 a + 28\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 + 3\cdot 29 + 25\cdot 29^{2} + 12\cdot 29^{3} + 9\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 12 a + 16 + \left(16 a + 16\right)\cdot 29 + \left(13 a + 7\right)\cdot 29^{2} + \left(23 a + 23\right)\cdot 29^{3} + \left(17 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 |
| $(1,2)(3,4)(5,6)$ |
| $(2,4)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,4)$ |
$-2$ |
| $3$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $3$ |
$2$ |
$(1,3)(4,6)$ |
$0$ |
| $2$ |
$3$ |
$(1,5,3)(2,4,6)$ |
$-1$ |
| $2$ |
$6$ |
$(1,4,5,6,3,2)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
