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:
| $r_{ 1 }$ |
$=$ |
$ 12 a + 10 + \left(3 a + 23\right)\cdot 29 + \left(27 a + 14\right)\cdot 29^{2} + \left(19 a + 11\right)\cdot 29^{3} + \left(28 a + 11\right)\cdot 29^{4} + \left(25 a + 23\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 29 + 5\cdot 29^{2} + 16\cdot 29^{3} + 2\cdot 29^{4} + 9\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 a + 26 + \left(18 a + 18\right)\cdot 29 + \left(19 a + 4\right)\cdot 29^{2} + \left(15 a + 4\right)\cdot 29^{3} + \left(10 a + 4\right)\cdot 29^{4} + \left(21 a + 22\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 a + 10 + \left(24 a + 5\right)\cdot 29 + a\cdot 29^{2} + \left(18 a + 24\right)\cdot 29^{3} + \left(25 a + 8\right)\cdot 29^{4} + 22\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 a + 26 + \left(4 a + 1\right)\cdot 29 + \left(27 a + 14\right)\cdot 29^{2} + \left(10 a + 25\right)\cdot 29^{3} + \left(3 a + 2\right)\cdot 29^{4} + \left(28 a + 1\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 a + 14 + \left(10 a + 9\right)\cdot 29 + \left(9 a + 26\right)\cdot 29^{2} + \left(13 a + 4\right)\cdot 29^{3} + \left(18 a + 12\right)\cdot 29^{4} + \left(7 a + 2\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 17 a + 12 + \left(25 a + 28\right)\cdot 29 + \left(a + 1\right)\cdot 29^{2} + \left(9 a + 26\right)\cdot 29^{3} + 18\cdot 29^{4} + \left(3 a + 8\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 15 + 27\cdot 29 + 19\cdot 29^{2} + 3\cdot 29^{3} + 26\cdot 29^{4} + 26\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,7,2)(3,6,5,4)$ |
| $(1,6,5)(3,7,4)$ |
| $(1,7)(2,8)(3,5)(4,6)$ |
| $(1,4)(3,5)(6,7)$ |
| $(1,5,7,3)(2,4,8,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,7)(2,8)(3,5)(4,6)$ |
$-4$ |
| $12$ |
$2$ |
$(1,4)(3,5)(6,7)$ |
$0$ |
| $8$ |
$3$ |
$(2,3,6)(4,8,5)$ |
$1$ |
| $6$ |
$4$ |
$(1,5,7,3)(2,4,8,6)$ |
$0$ |
| $8$ |
$6$ |
$(1,5,8,7,3,2)(4,6)$ |
$-1$ |
| $6$ |
$8$ |
$(1,8,6,3,7,2,4,5)$ |
$0$ |
| $6$ |
$8$ |
$(1,2,6,5,7,8,4,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
