Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 24 a + 8 + \left(15 a + 1\right)\cdot 29 + \left(6 a + 25\right)\cdot 29^{2} + \left(3 a + 18\right)\cdot 29^{3} + \left(26 a + 13\right)\cdot 29^{4} + \left(19 a + 5\right)\cdot 29^{5} + \left(4 a + 27\right)\cdot 29^{6} + \left(25 a + 16\right)\cdot 29^{7} + \left(14 a + 20\right)\cdot 29^{8} +O\left(29^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a + 9 + \left(13 a + 24\right)\cdot 29 + \left(6 a + 1\right)\cdot 29^{2} + \left(a + 14\right)\cdot 29^{3} + \left(3 a + 26\right)\cdot 29^{4} + \left(3 a + 20\right)\cdot 29^{5} + \left(23 a + 9\right)\cdot 29^{6} + \left(20 a + 18\right)\cdot 29^{7} + \left(25 a + 20\right)\cdot 29^{8} +O\left(29^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 + 20\cdot 29 + 19\cdot 29^{2} + 25\cdot 29^{3} + 14\cdot 29^{4} + 6\cdot 29^{5} + 3\cdot 29^{6} + 29^{7} + 8\cdot 29^{8} +O\left(29^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 16\cdot 29 + 6\cdot 29^{2} + 15\cdot 29^{3} + 24\cdot 29^{4} + 28\cdot 29^{5} + 9\cdot 29^{6} + 16\cdot 29^{7} + 13\cdot 29^{8} +O\left(29^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 a + 15 + \left(15 a + 25\right)\cdot 29 + \left(22 a + 20\right)\cdot 29^{2} + \left(27 a + 13\right)\cdot 29^{3} + \left(25 a + 11\right)\cdot 29^{4} + \left(25 a + 4\right)\cdot 29^{5} + \left(5 a + 6\right)\cdot 29^{6} + \left(8 a + 12\right)\cdot 29^{7} + \left(3 a + 12\right)\cdot 29^{8} +O\left(29^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 5 a + 12 + \left(13 a + 27\right)\cdot 29 + \left(22 a + 12\right)\cdot 29^{2} + \left(25 a + 28\right)\cdot 29^{3} + \left(2 a + 24\right)\cdot 29^{4} + \left(9 a + 20\right)\cdot 29^{5} + \left(24 a + 1\right)\cdot 29^{6} + \left(3 a + 22\right)\cdot 29^{7} + \left(14 a + 11\right)\cdot 29^{8} +O\left(29^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(4,6)$ |
| $(1,2,3)(4,6,5)$ |
| $(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,4)$ |
$-3$ |
| $3$ |
$2$ |
$(1,6)(3,4)$ |
$-1$ |
| $3$ |
$2$ |
$(1,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,3)(4,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,6)(2,3)(4,5)$ |
$-1$ |
| $8$ |
$3$ |
$(1,2,3)(4,6,5)$ |
$0$ |
| $6$ |
$4$ |
$(1,4,6,3)$ |
$1$ |
| $6$ |
$4$ |
$(1,4,6,3)(2,5)$ |
$-1$ |
| $8$ |
$6$ |
$(1,4,5,6,3,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
