Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 18 a + 6 + \left(22 a + 4\right)\cdot 29 + \left(10 a + 28\right)\cdot 29^{2} + \left(13 a + 13\right)\cdot 29^{4} + \left(2 a + 20\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a + 26 + \left(3 a + 3\right)\cdot 29 + \left(5 a + 16\right)\cdot 29^{2} + \left(18 a + 11\right)\cdot 29^{3} + \left(10 a + 28\right)\cdot 29^{4} + 9\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 + 21\cdot 29 + 14\cdot 29^{2} + 21\cdot 29^{3} + 29^{4} + 8\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 a + 12 + \left(25 a + 16\right)\cdot 29 + \left(23 a + 9\right)\cdot 29^{2} + \left(10 a + 10\right)\cdot 29^{3} + \left(18 a + 5\right)\cdot 29^{4} + \left(28 a + 1\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a + 15 + \left(9 a + 7\right)\cdot 29 + \left(2 a + 26\right)\cdot 29^{2} + \left(4 a + 24\right)\cdot 29^{3} + \left(19 a + 22\right)\cdot 29^{4} + \left(27 a + 12\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 a + 2 + \left(19 a + 16\right)\cdot 29 + \left(26 a + 28\right)\cdot 29^{2} + \left(24 a + 13\right)\cdot 29^{3} + \left(9 a + 27\right)\cdot 29^{4} + \left(a + 15\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 20 + 4\cdot 29 + 20\cdot 29^{2} + 11\cdot 29^{3} + 26\cdot 29^{4} + 27\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 11 a + 9 + \left(6 a + 12\right)\cdot 29 + \left(18 a + 1\right)\cdot 29^{2} + \left(28 a + 21\right)\cdot 29^{3} + \left(15 a + 19\right)\cdot 29^{4} + \left(26 a + 19\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,5,3)(6,7,8)$ |
| $(1,4)(2,8)(3,7)(5,6)$ |
| $(2,6)(3,7)(5,8)$ |
| $(1,2,4,8)(3,6,7,5)$ |
| $(1,5,4,6)(2,7,8,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,4)(2,8)(3,7)(5,6)$ |
$-4$ |
| $12$ |
$2$ |
$(2,6)(3,7)(5,8)$ |
$0$ |
| $8$ |
$3$ |
$(1,3,8)(2,4,7)$ |
$1$ |
| $6$ |
$4$ |
$(1,2,4,8)(3,6,7,5)$ |
$0$ |
| $8$ |
$6$ |
$(1,2,3,4,8,7)(5,6)$ |
$-1$ |
| $6$ |
$8$ |
$(1,2,7,6,4,8,3,5)$ |
$0$ |
| $6$ |
$8$ |
$(1,8,7,5,4,2,3,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
