Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 a + 14 + \left(17 a + 3\right)\cdot 29 + \left(27 a + 12\right)\cdot 29^{2} + \left(18 a + 24\right)\cdot 29^{3} + \left(25 a + 17\right)\cdot 29^{4} + \left(11 a + 26\right)\cdot 29^{5} + \left(16 a + 22\right)\cdot 29^{6} + \left(23 a + 21\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 23\cdot 29 + 17\cdot 29^{2} + 21\cdot 29^{3} + 27\cdot 29^{4} + 7\cdot 29^{5} + 22\cdot 29^{6} + 13\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ a + 5 + \left(5 a + 10\right)\cdot 29 + \left(22 a + 15\right)\cdot 29^{2} + \left(3 a + 3\right)\cdot 29^{3} + \left(18 a + 17\right)\cdot 29^{4} + 15\cdot 29^{5} + \left(11 a + 19\right)\cdot 29^{6} + \left(22 a + 10\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a + 19 + \left(5 a + 23\right)\cdot 29 + \left(22 a + 23\right)\cdot 29^{2} + \left(3 a + 28\right)\cdot 29^{3} + \left(18 a + 11\right)\cdot 29^{4} + 28\cdot 29^{5} + \left(11 a + 12\right)\cdot 29^{6} + \left(22 a + 4\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 23 a + 15 + \left(11 a + 25\right)\cdot 29 + \left(a + 16\right)\cdot 29^{2} + \left(10 a + 4\right)\cdot 29^{3} + \left(3 a + 11\right)\cdot 29^{4} + \left(17 a + 2\right)\cdot 29^{5} + \left(12 a + 6\right)\cdot 29^{6} + \left(5 a + 7\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 + 5\cdot 29 + 11\cdot 29^{2} + 7\cdot 29^{3} + 29^{4} + 21\cdot 29^{5} + 6\cdot 29^{6} + 15\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 28 a + 24 + \left(23 a + 18\right)\cdot 29 + \left(6 a + 13\right)\cdot 29^{2} + \left(25 a + 25\right)\cdot 29^{3} + \left(10 a + 11\right)\cdot 29^{4} + \left(28 a + 13\right)\cdot 29^{5} + \left(17 a + 9\right)\cdot 29^{6} + \left(6 a + 18\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 28 a + 10 + \left(23 a + 5\right)\cdot 29 + \left(6 a + 5\right)\cdot 29^{2} + 25 a\cdot 29^{3} + \left(10 a + 17\right)\cdot 29^{4} + 28 a\cdot 29^{5} + \left(17 a + 16\right)\cdot 29^{6} + \left(6 a + 24\right)\cdot 29^{7} +O\left(29^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,6)(2,5,8)$ |
| $(1,5)(2,4)(6,8)$ |
| $(1,3,5,7)(2,4,6,8)$ |
| $(1,8,5,4)(2,3,6,7)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$-2$ |
$-2$ |
| $12$ |
$2$ |
$(1,5)(2,4)(6,8)$ |
$0$ |
$0$ |
| $8$ |
$3$ |
$(2,4,7)(3,6,8)$ |
$-1$ |
$-1$ |
| $6$ |
$4$ |
$(1,8,5,4)(2,3,6,7)$ |
$0$ |
$0$ |
| $8$ |
$6$ |
$(1,5)(2,3,4,6,7,8)$ |
$1$ |
$1$ |
| $6$ |
$8$ |
$(1,6,7,4,5,2,3,8)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ |
$8$ |
$(1,2,7,8,5,6,3,4)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
