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:
| $r_{ 1 }$ |
$=$ |
$ 8\cdot 29 + 27\cdot 29^{3} + 22\cdot 29^{4} + 2\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a + 1 + \left(24 a + 19\right)\cdot 29 + 4\cdot 29^{2} + \left(10 a + 3\right)\cdot 29^{3} + \left(20 a + 1\right)\cdot 29^{4} + \left(6 a + 20\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 a + \left(7 a + 14\right)\cdot 29 + \left(9 a + 9\right)\cdot 29^{2} + \left(3 a + 23\right)\cdot 29^{3} + \left(21 a + 26\right)\cdot 29^{4} + \left(26 a + 9\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 a + 4 + 21 a\cdot 29 + \left(19 a + 19\right)\cdot 29^{2} + \left(25 a + 1\right)\cdot 29^{3} + \left(7 a + 13\right)\cdot 29^{4} + \left(2 a + 6\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 + 22\cdot 29 + 18\cdot 29^{2} + 19\cdot 29^{3} + 7\cdot 29^{4} + 25\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 a + 3 + \left(3 a + 20\right)\cdot 29 + \left(4 a + 16\right)\cdot 29^{2} + \left(2 a + 5\right)\cdot 29^{3} + \left(17 a + 21\right)\cdot 29^{4} + \left(23 a + 16\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 26 a + 16 + \left(4 a + 20\right)\cdot 29 + \left(28 a + 13\right)\cdot 29^{2} + \left(18 a + 23\right)\cdot 29^{3} + \left(8 a + 5\right)\cdot 29^{4} + \left(22 a + 4\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ a + 27 + \left(25 a + 11\right)\cdot 29 + \left(24 a + 4\right)\cdot 29^{2} + \left(26 a + 12\right)\cdot 29^{3} + \left(11 a + 17\right)\cdot 29^{4} + \left(5 a + 1\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,5,4)(2,3,7,6)$ |
| $(1,4,3)(5,8,6)$ |
| $(1,3,5,6)(2,4,7,8)$ |
| $(1,5)(3,8)(4,6)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,7)(3,6)(4,8)$ |
$-2$ |
$-2$ |
| $12$ |
$2$ |
$(1,5)(3,8)(4,6)$ |
$0$ |
$0$ |
| $8$ |
$3$ |
$(1,4,3)(5,8,6)$ |
$-1$ |
$-1$ |
| $6$ |
$4$ |
$(1,3,5,6)(2,4,7,8)$ |
$0$ |
$0$ |
| $8$ |
$6$ |
$(1,8,3,5,4,6)(2,7)$ |
$1$ |
$1$ |
| $6$ |
$8$ |
$(1,6,7,8,5,3,2,4)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ |
$8$ |
$(1,3,7,4,5,6,2,8)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
