Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 25.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 a + 2 + \left(a + 26\right)\cdot 29 + \left(2 a + 10\right)\cdot 29^{2} + \left(18 a + 15\right)\cdot 29^{3} + \left(23 a + 12\right)\cdot 29^{4} + \left(12 a + 10\right)\cdot 29^{5} + \left(28 a + 19\right)\cdot 29^{6} + \left(2 a + 1\right)\cdot 29^{7} + \left(2 a + 22\right)\cdot 29^{8} + \left(19 a + 9\right)\cdot 29^{9} + \left(12 a + 12\right)\cdot 29^{10} + 6 a\cdot 29^{11} + \left(5 a + 4\right)\cdot 29^{12} + \left(23 a + 8\right)\cdot 29^{13} + 24\cdot 29^{14} + \left(8 a + 25\right)\cdot 29^{15} + \left(9 a + 22\right)\cdot 29^{16} + \left(5 a + 20\right)\cdot 29^{17} + \left(24 a + 15\right)\cdot 29^{18} + \left(24 a + 26\right)\cdot 29^{19} + \left(4 a + 15\right)\cdot 29^{20} + \left(13 a + 13\right)\cdot 29^{21} + \left(23 a + 11\right)\cdot 29^{22} + 25\cdot 29^{23} + \left(4 a + 4\right)\cdot 29^{24} +O\left(29^{ 25 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 20\cdot 29 + 8\cdot 29^{2} + 8\cdot 29^{3} + 25\cdot 29^{4} + 22\cdot 29^{5} + 21\cdot 29^{6} + 4\cdot 29^{7} + 6\cdot 29^{9} + 2\cdot 29^{10} + 26\cdot 29^{11} + 12\cdot 29^{12} + 13\cdot 29^{13} + 14\cdot 29^{14} + 19\cdot 29^{15} + 24\cdot 29^{16} + 10\cdot 29^{17} + 29^{18} + 21\cdot 29^{19} + 11\cdot 29^{20} + 2\cdot 29^{21} + 9\cdot 29^{22} + 8\cdot 29^{23} + 18\cdot 29^{24} +O\left(29^{ 25 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 a + 16 + \left(15 a + 13\right)\cdot 29 + \left(3 a + 22\right)\cdot 29^{2} + \left(20 a + 22\right)\cdot 29^{3} + \left(20 a + 20\right)\cdot 29^{4} + 3 a\cdot 29^{5} + \left(27 a + 3\right)\cdot 29^{6} + \left(27 a + 18\right)\cdot 29^{7} + \left(14 a + 3\right)\cdot 29^{8} + \left(18 a + 5\right)\cdot 29^{9} + \left(11 a + 25\right)\cdot 29^{10} + 16\cdot 29^{11} + \left(23 a + 3\right)\cdot 29^{12} + \left(25 a + 5\right)\cdot 29^{13} + \left(16 a + 12\right)\cdot 29^{14} + \left(20 a + 4\right)\cdot 29^{15} + \left(2 a + 12\right)\cdot 29^{16} + \left(28 a + 3\right)\cdot 29^{17} + \left(10 a + 10\right)\cdot 29^{18} + \left(12 a + 14\right)\cdot 29^{19} + \left(9 a + 20\right)\cdot 29^{20} + 10\cdot 29^{21} + \left(24 a + 19\right)\cdot 29^{22} + \left(8 a + 26\right)\cdot 29^{23} + \left(4 a + 25\right)\cdot 29^{24} +O\left(29^{ 25 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 a + 28 + \left(13 a + 18\right)\cdot 29 + \left(25 a + 24\right)\cdot 29^{2} + \left(8 a + 3\right)\cdot 29^{3} + \left(8 a + 17\right)\cdot 29^{4} + \left(25 a + 27\right)\cdot 29^{5} + \left(a + 18\right)\cdot 29^{6} + \left(a + 14\right)\cdot 29^{7} + \left(14 a + 21\right)\cdot 29^{8} + \left(10 a + 24\right)\cdot 29^{9} + \left(17 a + 6\right)\cdot 29^{10} + \left(28 a + 7\right)\cdot 29^{11} + \left(5 a + 2\right)\cdot 29^{12} + \left(3 a + 24\right)\cdot 29^{13} + \left(12 a + 12\right)\cdot 29^{14} + \left(8 a + 3\right)\cdot 29^{15} + \left(26 a + 5\right)\cdot 29^{16} + 25\cdot 29^{17} + \left(18 a + 7\right)\cdot 29^{18} + \left(16 a + 7\right)\cdot 29^{19} + \left(19 a + 26\right)\cdot 29^{20} + \left(28 a + 2\right)\cdot 29^{21} + \left(4 a + 23\right)\cdot 29^{22} + \left(20 a + 17\right)\cdot 29^{23} + \left(24 a + 9\right)\cdot 29^{24} +O\left(29^{ 25 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 a + \left(18 a + 10\right)\cdot 29 + \left(24 a + 25\right)\cdot 29^{2} + \left(26 a + 14\right)\cdot 29^{3} + \left(18 a + 11\right)\cdot 29^{4} + \left(19 a + 21\right)\cdot 29^{5} + \left(2 a + 7\right)\cdot 29^{6} + \left(25 a + 3\right)\cdot 29^{7} + \left(6 a + 4\right)\cdot 29^{8} + \left(7 a + 5\right)\cdot 29^{9} + \left(9 a + 4\right)\cdot 29^{10} + \left(3 a + 10\right)\cdot 29^{11} + \left(4 a + 10\right)\cdot 29^{12} + \left(19 a + 14\right)\cdot 29^{13} + \left(26 a + 16\right)\cdot 29^{14} + \left(25 a + 12\right)\cdot 29^{15} + \left(12 a + 6\right)\cdot 29^{16} + \left(3 a + 12\right)\cdot 29^{17} + \left(9 a + 6\right)\cdot 29^{18} + \left(a + 1\right)\cdot 29^{19} + \left(17 a + 20\right)\cdot 29^{20} + \left(17 a + 23\right)\cdot 29^{21} + \left(11 a + 7\right)\cdot 29^{22} + \left(17 a + 8\right)\cdot 29^{23} + \left(16 a + 8\right)\cdot 29^{24} +O\left(29^{ 25 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 16 a + 9 + \left(27 a + 20\right)\cdot 29 + \left(26 a + 19\right)\cdot 29^{2} + \left(10 a + 16\right)\cdot 29^{3} + \left(5 a + 25\right)\cdot 29^{4} + \left(16 a + 21\right)\cdot 29^{5} + 3\cdot 29^{6} + \left(26 a + 17\right)\cdot 29^{7} + 26 a\cdot 29^{8} + \left(9 a + 16\right)\cdot 29^{9} + \left(16 a + 27\right)\cdot 29^{10} + \left(22 a + 19\right)\cdot 29^{11} + \left(23 a + 23\right)\cdot 29^{12} + \left(5 a + 2\right)\cdot 29^{13} + \left(28 a + 5\right)\cdot 29^{14} + \left(20 a + 7\right)\cdot 29^{15} + \left(19 a + 3\right)\cdot 29^{16} + \left(23 a + 9\right)\cdot 29^{17} + \left(4 a + 15\right)\cdot 29^{18} + \left(4 a + 10\right)\cdot 29^{19} + \left(24 a + 15\right)\cdot 29^{20} + \left(15 a + 16\right)\cdot 29^{21} + \left(5 a + 28\right)\cdot 29^{22} + \left(28 a + 5\right)\cdot 29^{23} + \left(24 a + 24\right)\cdot 29^{24} +O\left(29^{ 25 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 18 a + 26 + \left(10 a + 3\right)\cdot 29 + \left(4 a + 14\right)\cdot 29^{2} + \left(2 a + 8\right)\cdot 29^{3} + \left(10 a + 21\right)\cdot 29^{4} + \left(9 a + 13\right)\cdot 29^{5} + \left(26 a + 1\right)\cdot 29^{6} + \left(3 a + 10\right)\cdot 29^{7} + \left(22 a + 13\right)\cdot 29^{8} + \left(21 a + 5\right)\cdot 29^{9} + \left(19 a + 14\right)\cdot 29^{10} + \left(25 a + 17\right)\cdot 29^{11} + \left(24 a + 27\right)\cdot 29^{12} + \left(9 a + 18\right)\cdot 29^{13} + \left(2 a + 14\right)\cdot 29^{14} + \left(3 a + 28\right)\cdot 29^{15} + \left(16 a + 15\right)\cdot 29^{16} + \left(25 a + 16\right)\cdot 29^{17} + \left(19 a + 19\right)\cdot 29^{18} + \left(27 a + 27\right)\cdot 29^{19} + \left(11 a + 16\right)\cdot 29^{20} + \left(11 a + 7\right)\cdot 29^{21} + \left(17 a + 19\right)\cdot 29^{22} + \left(11 a + 25\right)\cdot 29^{23} + \left(12 a + 15\right)\cdot 29^{24} +O\left(29^{ 25 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 15 + 2\cdot 29 + 19\cdot 29^{2} + 25\cdot 29^{3} + 10\cdot 29^{4} + 26\cdot 29^{5} + 10\cdot 29^{6} + 17\cdot 29^{7} + 21\cdot 29^{8} + 14\cdot 29^{9} + 23\cdot 29^{10} + 17\cdot 29^{11} + 2\cdot 29^{12} + 16\cdot 29^{14} + 14\cdot 29^{15} + 25\cdot 29^{16} + 17\cdot 29^{17} + 10\cdot 29^{18} + 7\cdot 29^{19} + 18\cdot 29^{20} + 9\cdot 29^{21} + 26\cdot 29^{22} + 26\cdot 29^{23} + 8\cdot 29^{24} +O\left(29^{ 25 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,8)(3,5)$ |
| $(3,5)(4,7)$ |
| $(2,5,7)(3,4,8)$ |
| $(1,3)(2,8)(5,6)$ |
| $(1,5,6,3)(2,7,8,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,6)(2,8)(3,5)(4,7)$ |
$-4$ |
| $6$ |
$2$ |
$(3,5)(4,7)$ |
$0$ |
| $12$ |
$2$ |
$(1,5)(2,7)(3,6)(4,8)$ |
$0$ |
| $24$ |
$2$ |
$(1,3)(2,8)(5,6)$ |
$0$ |
| $32$ |
$3$ |
$(1,7,3)(4,5,6)$ |
$1$ |
| $6$ |
$4$ |
$(1,5,6,3)(2,7,8,4)$ |
$0$ |
| $6$ |
$4$ |
$(1,7,6,4)(2,5,8,3)$ |
$0$ |
| $12$ |
$4$ |
$(1,3,6,5)(2,8)(4,7)$ |
$2$ |
| $12$ |
$4$ |
$(1,4,6,7)$ |
$-2$ |
| $32$ |
$6$ |
$(1,2,5,6,8,3)(4,7)$ |
$-1$ |
| $24$ |
$8$ |
$(1,4,8,5,6,7,2,3)$ |
$0$ |
| $24$ |
$8$ |
$(1,8,7,3,6,2,4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
