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:
| $r_{ 1 }$ |
$=$ |
$ 18 a + 24 + \left(15 a + 22\right)\cdot 29 + \left(15 a + 1\right)\cdot 29^{2} + \left(24 a + 17\right)\cdot 29^{3} + \left(13 a + 2\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 a + 4 + \left(10 a + 28\right)\cdot 29 + \left(6 a + 23\right)\cdot 29^{2} + \left(3 a + 4\right)\cdot 29^{3} + \left(25 a + 1\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a + 27 + \left(13 a + 24\right)\cdot 29 + \left(13 a + 5\right)\cdot 29^{2} + \left(4 a + 8\right)\cdot 29^{3} + \left(15 a + 18\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 + 3\cdot 29 + 10\cdot 29^{2} + 13\cdot 29^{3} + 28\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 a + 21 + \left(18 a + 7\right)\cdot 29 + \left(22 a + 16\right)\cdot 29^{2} + \left(25 a + 14\right)\cdot 29^{3} + \left(3 a + 7\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $10$ |
$2$ |
$(1,2)$ |
$-1$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $30$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $24$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $20$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
