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:
| $r_{ 1 }$ |
$=$ |
$ 19 a + \left(25 a + 21\right)\cdot 29 + \left(25 a + 10\right)\cdot 29^{2} + \left(7 a + 4\right)\cdot 29^{3} + \left(12 a + 4\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 a + 8 + \left(3 a + 14\right)\cdot 29 + \left(3 a + 27\right)\cdot 29^{2} + \left(21 a + 17\right)\cdot 29^{3} + \left(16 a + 28\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 + 24\cdot 29 + 2\cdot 29^{2} + 8\cdot 29^{3} + 18\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 a + 14 + \left(16 a + 17\right)\cdot 29 + \left(23 a + 20\right)\cdot 29^{2} + \left(11 a + 21\right)\cdot 29^{3} + \left(6 a + 9\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ a + 9 + \left(12 a + 16\right)\cdot 29 + \left(5 a + 5\right)\cdot 29^{2} + \left(17 a + 28\right)\cdot 29^{3} + 22 a\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 + 22\cdot 29 + 19\cdot 29^{2} + 6\cdot 29^{3} + 25\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(3,4)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$0$ |
| $6$ |
$2$ |
$(3,4)$ |
$2$ |
| $9$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,6)(3,4,5)$ |
$-2$ |
| $4$ |
$3$ |
$(1,2,6)$ |
$1$ |
| $18$ |
$4$ |
$(1,3,2,4)(5,6)$ |
$0$ |
| $12$ |
$6$ |
$(1,4,2,5,6,3)$ |
$0$ |
| $12$ |
$6$ |
$(1,2,6)(3,4)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
