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