Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 a + 28 + \left(9 a + 25\right)\cdot 29 + \left(13 a + 14\right)\cdot 29^{2} + 2 a\cdot 29^{3} + \left(11 a + 17\right)\cdot 29^{4} + \left(4 a + 23\right)\cdot 29^{5} + \left(a + 13\right)\cdot 29^{6} + \left(13 a + 11\right)\cdot 29^{7} + \left(24 a + 3\right)\cdot 29^{8} + \left(15 a + 16\right)\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 24\cdot 29 + 17\cdot 29^{2} + 13\cdot 29^{3} + 18\cdot 29^{4} + 11\cdot 29^{5} + 17\cdot 29^{6} + 6\cdot 29^{7} + 7\cdot 29^{8} + 20\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ a + 26 + \left(13 a + 2\right)\cdot 29 + \left(24 a + 19\right)\cdot 29^{2} + \left(21 a + 5\right)\cdot 29^{3} + 14\cdot 29^{4} + \left(12 a + 3\right)\cdot 29^{5} + 18 a\cdot 29^{6} + \left(21 a + 1\right)\cdot 29^{7} + \left(9 a + 19\right)\cdot 29^{8} + \left(24 a + 7\right)\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 a + 2 + \left(15 a + 9\right)\cdot 29 + \left(4 a + 12\right)\cdot 29^{2} + \left(7 a + 3\right)\cdot 29^{3} + \left(28 a + 25\right)\cdot 29^{4} + \left(16 a + 4\right)\cdot 29^{5} + \left(10 a + 22\right)\cdot 29^{6} + \left(7 a + 3\right)\cdot 29^{7} + \left(19 a + 17\right)\cdot 29^{8} + \left(4 a + 3\right)\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 a + 1 + \left(19 a + 3\right)\cdot 29 + \left(15 a + 14\right)\cdot 29^{2} + \left(26 a + 28\right)\cdot 29^{3} + \left(17 a + 11\right)\cdot 29^{4} + \left(24 a + 5\right)\cdot 29^{5} + \left(27 a + 15\right)\cdot 29^{6} + \left(15 a + 17\right)\cdot 29^{7} + \left(4 a + 25\right)\cdot 29^{8} + \left(13 a + 12\right)\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 26 + 4\cdot 29 + 11\cdot 29^{2} + 15\cdot 29^{3} + 10\cdot 29^{4} + 17\cdot 29^{5} + 11\cdot 29^{6} + 22\cdot 29^{7} + 21\cdot 29^{8} + 8\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 28 a + 3 + \left(15 a + 26\right)\cdot 29 + \left(4 a + 9\right)\cdot 29^{2} + \left(7 a + 23\right)\cdot 29^{3} + \left(28 a + 14\right)\cdot 29^{4} + \left(16 a + 25\right)\cdot 29^{5} + \left(10 a + 28\right)\cdot 29^{6} + \left(7 a + 27\right)\cdot 29^{7} + \left(19 a + 9\right)\cdot 29^{8} + \left(4 a + 21\right)\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ a + 27 + \left(13 a + 19\right)\cdot 29 + \left(24 a + 16\right)\cdot 29^{2} + \left(21 a + 25\right)\cdot 29^{3} + 3\cdot 29^{4} + \left(12 a + 24\right)\cdot 29^{5} + \left(18 a + 6\right)\cdot 29^{6} + \left(21 a + 25\right)\cdot 29^{7} + \left(9 a + 11\right)\cdot 29^{8} + \left(24 a + 25\right)\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,7)(2,3,5)$ |
| $(1,5)(2,7)(3,6)$ |
| $(1,7,5,3)(2,8,6,4)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
| $(1,6,5,2)(3,4,7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$-2$ |
$-2$ |
| $12$ |
$2$ |
$(1,5)(2,7)(3,6)$ |
$0$ |
$0$ |
| $8$ |
$3$ |
$(1,4,2)(5,8,6)$ |
$-1$ |
$-1$ |
| $6$ |
$4$ |
$(1,7,5,3)(2,8,6,4)$ |
$0$ |
$0$ |
| $8$ |
$6$ |
$(1,6,4,5,2,8)(3,7)$ |
$1$ |
$1$ |
| $6$ |
$8$ |
$(1,3,4,2,5,7,8,6)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ |
$8$ |
$(1,7,4,6,5,3,8,2)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
