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