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 a + 9 + \left(17 a + 9\right)\cdot 29 + \left(27 a + 4\right)\cdot 29^{2} + 25 a\cdot 29^{3} + \left(9 a + 18\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 20\cdot 29 + 26\cdot 29^{2} + 24\cdot 29^{3} + 25\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a + 17 + \left(25 a + 1\right)\cdot 29 + \left(3 a + 11\right)\cdot 29^{2} + 12 a\cdot 29^{3} + \left(11 a + 22\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 a + 20 + \left(3 a + 24\right)\cdot 29 + \left(25 a + 4\right)\cdot 29^{2} + \left(16 a + 28\right)\cdot 29^{3} + \left(17 a + 8\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 + 26\cdot 29 + 29^{2} + 18\cdot 29^{3} + 28\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 26 a + 24 + \left(11 a + 4\right)\cdot 29 + \left(a + 9\right)\cdot 29^{2} + \left(3 a + 15\right)\cdot 29^{3} + \left(19 a + 12\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)$ |
| $(2,5)$ |
| $(1,2,3)(4,6,5)$ |
| $(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,4)$ |
$-3$ |
| $3$ |
$2$ |
$(3,4)$ |
$1$ |
| $3$ |
$2$ |
$(1,6)(3,4)$ |
$-1$ |
| $4$ |
$3$ |
$(1,2,3)(4,6,5)$ |
$0$ |
| $4$ |
$3$ |
$(1,3,2)(4,5,6)$ |
$0$ |
| $4$ |
$6$ |
$(1,2,3,6,5,4)$ |
$0$ |
| $4$ |
$6$ |
$(1,4,5,6,3,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
