Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 85.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 a + 11 + \left(6 a + 5\right)\cdot 29 + \left(11 a + 16\right)\cdot 29^{2} + \left(2 a + 14\right)\cdot 29^{3} + \left(a + 12\right)\cdot 29^{4} + \left(18 a + 8\right)\cdot 29^{5} + \left(3 a + 12\right)\cdot 29^{6} + \left(9 a + 23\right)\cdot 29^{7} + 28 a\cdot 29^{8} + \left(3 a + 3\right)\cdot 29^{9} + \left(19 a + 12\right)\cdot 29^{10} + \left(8 a + 15\right)\cdot 29^{11} + \left(21 a + 17\right)\cdot 29^{12} + \left(20 a + 25\right)\cdot 29^{13} + \left(22 a + 24\right)\cdot 29^{14} + \left(9 a + 21\right)\cdot 29^{15} + \left(18 a + 2\right)\cdot 29^{16} + \left(11 a + 24\right)\cdot 29^{17} + \left(27 a + 3\right)\cdot 29^{18} + \left(5 a + 7\right)\cdot 29^{19} + \left(3 a + 15\right)\cdot 29^{20} + \left(28 a + 2\right)\cdot 29^{21} + \left(15 a + 28\right)\cdot 29^{22} + \left(7 a + 19\right)\cdot 29^{23} + \left(8 a + 11\right)\cdot 29^{24} + \left(28 a + 12\right)\cdot 29^{25} + \left(28 a + 27\right)\cdot 29^{26} + 18 a\cdot 29^{27} + 21\cdot 29^{28} + \left(4 a + 19\right)\cdot 29^{29} + \left(18 a + 7\right)\cdot 29^{30} + \left(28 a + 17\right)\cdot 29^{31} + \left(26 a + 2\right)\cdot 29^{32} + \left(6 a + 24\right)\cdot 29^{33} + \left(15 a + 11\right)\cdot 29^{34} + \left(21 a + 15\right)\cdot 29^{35} + \left(19 a + 12\right)\cdot 29^{36} + \left(6 a + 15\right)\cdot 29^{37} + \left(24 a + 27\right)\cdot 29^{38} + \left(8 a + 24\right)\cdot 29^{39} + \left(14 a + 4\right)\cdot 29^{40} + \left(8 a + 17\right)\cdot 29^{41} + \left(16 a + 25\right)\cdot 29^{42} + \left(4 a + 23\right)\cdot 29^{43} + \left(26 a + 3\right)\cdot 29^{44} + \left(24 a + 19\right)\cdot 29^{45} + 20\cdot 29^{46} + \left(2 a + 25\right)\cdot 29^{47} + \left(11 a + 2\right)\cdot 29^{48} + \left(14 a + 4\right)\cdot 29^{49} + \left(13 a + 7\right)\cdot 29^{50} + \left(14 a + 26\right)\cdot 29^{51} + \left(20 a + 3\right)\cdot 29^{52} + \left(16 a + 14\right)\cdot 29^{53} + \left(27 a + 24\right)\cdot 29^{54} + \left(26 a + 24\right)\cdot 29^{55} + \left(7 a + 3\right)\cdot 29^{56} + \left(14 a + 7\right)\cdot 29^{57} + \left(6 a + 2\right)\cdot 29^{58} + \left(17 a + 28\right)\cdot 29^{59} + \left(23 a + 17\right)\cdot 29^{60} + \left(4 a + 26\right)\cdot 29^{61} + \left(12 a + 13\right)\cdot 29^{62} + \left(15 a + 11\right)\cdot 29^{63} + \left(24 a + 3\right)\cdot 29^{64} + \left(18 a + 16\right)\cdot 29^{65} + \left(24 a + 8\right)\cdot 29^{66} + \left(24 a + 9\right)\cdot 29^{67} + \left(8 a + 16\right)\cdot 29^{68} + 19\cdot 29^{69} + \left(4 a + 5\right)\cdot 29^{70} + \left(13 a + 12\right)\cdot 29^{71} + \left(19 a + 22\right)\cdot 29^{72} + \left(17 a + 26\right)\cdot 29^{73} + \left(18 a + 10\right)\cdot 29^{74} + \left(2 a + 20\right)\cdot 29^{75} + \left(4 a + 21\right)\cdot 29^{76} + \left(5 a + 9\right)\cdot 29^{77} + \left(7 a + 3\right)\cdot 29^{78} + \left(8 a + 26\right)\cdot 29^{79} + \left(13 a + 16\right)\cdot 29^{80} + \left(16 a + 28\right)\cdot 29^{81} + \left(a + 9\right)\cdot 29^{82} + \left(17 a + 17\right)\cdot 29^{83} + \left(12 a + 18\right)\cdot 29^{84} +O\left(29^{ 85 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 a + 24 + \left(10 a + 17\right)\cdot 29 + \left(12 a + 21\right)\cdot 29^{2} + \left(17 a + 4\right)\cdot 29^{3} + \left(8 a + 23\right)\cdot 29^{4} + \left(17 a + 1\right)\cdot 29^{5} + \left(5 a + 3\right)\cdot 29^{6} + \left(14 a + 21\right)\cdot 29^{7} + \left(12 a + 22\right)\cdot 29^{8} + \left(13 a + 27\right)\cdot 29^{9} + \left(13 a + 20\right)\cdot 29^{10} + \left(16 a + 28\right)\cdot 29^{11} + \left(27 a + 3\right)\cdot 29^{12} + \left(5 a + 24\right)\cdot 29^{13} + \left(20 a + 14\right)\cdot 29^{14} + \left(6 a + 6\right)\cdot 29^{15} + \left(4 a + 13\right)\cdot 29^{16} + \left(10 a + 2\right)\cdot 29^{17} + \left(22 a + 5\right)\cdot 29^{18} + \left(15 a + 2\right)\cdot 29^{19} + \left(20 a + 1\right)\cdot 29^{20} + \left(26 a + 6\right)\cdot 29^{21} + \left(25 a + 7\right)\cdot 29^{22} + \left(18 a + 11\right)\cdot 29^{23} + \left(25 a + 2\right)\cdot 29^{24} + \left(26 a + 21\right)\cdot 29^{25} + \left(4 a + 25\right)\cdot 29^{26} + \left(7 a + 16\right)\cdot 29^{27} + \left(13 a + 3\right)\cdot 29^{28} + \left(5 a + 14\right)\cdot 29^{29} + \left(20 a + 26\right)\cdot 29^{30} + \left(3 a + 7\right)\cdot 29^{31} + \left(13 a + 7\right)\cdot 29^{32} + \left(9 a + 11\right)\cdot 29^{33} + \left(24 a + 17\right)\cdot 29^{34} + \left(10 a + 21\right)\cdot 29^{35} + \left(6 a + 1\right)\cdot 29^{36} + \left(28 a + 6\right)\cdot 29^{37} + \left(24 a + 5\right)\cdot 29^{38} + \left(9 a + 8\right)\cdot 29^{39} + \left(27 a + 21\right)\cdot 29^{40} + \left(28 a + 24\right)\cdot 29^{41} + \left(5 a + 4\right)\cdot 29^{42} + \left(7 a + 26\right)\cdot 29^{43} + \left(a + 8\right)\cdot 29^{44} + \left(27 a + 13\right)\cdot 29^{45} + \left(24 a + 1\right)\cdot 29^{46} + \left(15 a + 13\right)\cdot 29^{47} + \left(6 a + 12\right)\cdot 29^{48} + \left(16 a + 2\right)\cdot 29^{49} + \left(11 a + 6\right)\cdot 29^{50} + 3\cdot 29^{51} + \left(7 a + 18\right)\cdot 29^{52} + \left(27 a + 1\right)\cdot 29^{53} + \left(5 a + 15\right)\cdot 29^{54} + \left(11 a + 13\right)\cdot 29^{55} + 9 a\cdot 29^{56} + \left(12 a + 28\right)\cdot 29^{57} + \left(18 a + 25\right)\cdot 29^{58} + \left(11 a + 8\right)\cdot 29^{59} + \left(6 a + 23\right)\cdot 29^{60} + \left(10 a + 14\right)\cdot 29^{61} + \left(19 a + 28\right)\cdot 29^{62} + \left(23 a + 13\right)\cdot 29^{63} + \left(4 a + 11\right)\cdot 29^{64} + \left(28 a + 22\right)\cdot 29^{65} + \left(14 a + 23\right)\cdot 29^{66} + \left(19 a + 26\right)\cdot 29^{67} + \left(28 a + 9\right)\cdot 29^{68} + \left(21 a + 17\right)\cdot 29^{69} + \left(18 a + 13\right)\cdot 29^{70} + 20\cdot 29^{71} + \left(10 a + 28\right)\cdot 29^{72} + \left(4 a + 6\right)\cdot 29^{73} + \left(26 a + 15\right)\cdot 29^{74} + \left(a + 12\right)\cdot 29^{75} + \left(24 a + 26\right)\cdot 29^{76} + \left(a + 11\right)\cdot 29^{77} + \left(15 a + 5\right)\cdot 29^{78} + \left(20 a + 24\right)\cdot 29^{79} + \left(a + 6\right)\cdot 29^{80} + \left(24 a + 13\right)\cdot 29^{81} + \left(2 a + 15\right)\cdot 29^{82} + \left(21 a + 21\right)\cdot 29^{83} + \left(28 a + 20\right)\cdot 29^{84} +O\left(29^{ 85 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 a + 10 + \left(25 a + 21\right)\cdot 29 + \left(19 a + 25\right)\cdot 29^{2} + \left(25 a + 2\right)\cdot 29^{3} + \left(7 a + 1\right)\cdot 29^{4} + \left(22 a + 12\right)\cdot 29^{5} + \left(5 a + 27\right)\cdot 29^{6} + \left(7 a + 12\right)\cdot 29^{7} + \left(19 a + 22\right)\cdot 29^{8} + \left(14 a + 7\right)\cdot 29^{9} + \left(4 a + 27\right)\cdot 29^{10} + \left(15 a + 20\right)\cdot 29^{11} + \left(23 a + 9\right)\cdot 29^{12} + \left(22 a + 10\right)\cdot 29^{13} + \left(21 a + 16\right)\cdot 29^{14} + \left(15 a + 28\right)\cdot 29^{15} + \left(18 a + 10\right)\cdot 29^{16} + \left(23 a + 7\right)\cdot 29^{17} + \left(20 a + 7\right)\cdot 29^{18} + 23 a\cdot 29^{19} + \left(18 a + 3\right)\cdot 29^{20} + \left(26 a + 17\right)\cdot 29^{21} + \left(7 a + 28\right)\cdot 29^{22} + \left(18 a + 15\right)\cdot 29^{23} + \left(9 a + 6\right)\cdot 29^{24} + \left(19 a + 27\right)\cdot 29^{25} + \left(13 a + 5\right)\cdot 29^{26} + \left(28 a + 17\right)\cdot 29^{27} + \left(22 a + 2\right)\cdot 29^{28} + \left(21 a + 24\right)\cdot 29^{29} + \left(8 a + 12\right)\cdot 29^{30} + \left(4 a + 21\right)\cdot 29^{31} + \left(28 a + 1\right)\cdot 29^{32} + \left(5 a + 16\right)\cdot 29^{33} + \left(7 a + 4\right)\cdot 29^{34} + \left(20 a + 25\right)\cdot 29^{35} + \left(17 a + 3\right)\cdot 29^{36} + \left(28 a + 18\right)\cdot 29^{37} + \left(26 a + 6\right)\cdot 29^{38} + \left(9 a + 9\right)\cdot 29^{39} + \left(19 a + 5\right)\cdot 29^{40} + \left(13 a + 16\right)\cdot 29^{41} + \left(9 a + 9\right)\cdot 29^{42} + \left(19 a + 9\right)\cdot 29^{43} + \left(20 a + 13\right)\cdot 29^{44} + \left(22 a + 2\right)\cdot 29^{45} + \left(18 a + 12\right)\cdot 29^{46} + \left(10 a + 17\right)\cdot 29^{47} + \left(23 a + 8\right)\cdot 29^{48} + \left(14 a + 1\right)\cdot 29^{49} + \left(4 a + 28\right)\cdot 29^{50} + \left(28 a + 9\right)\cdot 29^{51} + \left(7 a + 21\right)\cdot 29^{52} + \left(7 a + 4\right)\cdot 29^{53} + \left(23 a + 18\right)\cdot 29^{54} + \left(10 a + 7\right)\cdot 29^{55} + \left(20 a + 25\right)\cdot 29^{56} + \left(15 a + 13\right)\cdot 29^{57} + \left(21 a + 28\right)\cdot 29^{58} + 19\cdot 29^{59} + \left(10 a + 28\right)\cdot 29^{60} + \left(22 a + 12\right)\cdot 29^{61} + 20 a\cdot 29^{62} + \left(26 a + 2\right)\cdot 29^{63} + \left(18 a + 21\right)\cdot 29^{64} + \left(8 a + 18\right)\cdot 29^{65} + \left(19 a + 25\right)\cdot 29^{66} + \left(7 a + 1\right)\cdot 29^{67} + \left(15 a + 7\right)\cdot 29^{68} + \left(14 a + 2\right)\cdot 29^{69} + \left(4 a + 4\right)\cdot 29^{70} + \left(23 a + 23\right)\cdot 29^{71} + \left(26 a + 15\right)\cdot 29^{72} + \left(18 a + 1\right)\cdot 29^{73} + \left(4 a + 18\right)\cdot 29^{74} + \left(9 a + 26\right)\cdot 29^{75} + \left(25 a + 16\right)\cdot 29^{76} + \left(7 a + 19\right)\cdot 29^{77} + \left(27 a + 23\right)\cdot 29^{78} + \left(5 a + 15\right)\cdot 29^{79} + \left(5 a + 25\right)\cdot 29^{80} + \left(23 a + 18\right)\cdot 29^{81} + \left(6 a + 4\right)\cdot 29^{82} + \left(22 a + 27\right)\cdot 29^{83} + \left(23 a + 9\right)\cdot 29^{84} +O\left(29^{ 85 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 a + 1 + \left(3 a + 27\right)\cdot 29 + \left(9 a + 12\right)\cdot 29^{2} + \left(3 a + 24\right)\cdot 29^{3} + \left(21 a + 14\right)\cdot 29^{4} + \left(6 a + 28\right)\cdot 29^{5} + \left(23 a + 4\right)\cdot 29^{6} + \left(21 a + 14\right)\cdot 29^{7} + \left(9 a + 24\right)\cdot 29^{8} + \left(14 a + 3\right)\cdot 29^{9} + \left(24 a + 6\right)\cdot 29^{10} + \left(13 a + 5\right)\cdot 29^{11} + \left(5 a + 25\right)\cdot 29^{12} + \left(6 a + 13\right)\cdot 29^{13} + \left(7 a + 15\right)\cdot 29^{14} + \left(13 a + 27\right)\cdot 29^{15} + 10 a\cdot 29^{16} + \left(5 a + 20\right)\cdot 29^{17} + 8 a\cdot 29^{18} + \left(5 a + 11\right)\cdot 29^{19} + \left(10 a + 15\right)\cdot 29^{20} + \left(2 a + 15\right)\cdot 29^{21} + \left(21 a + 12\right)\cdot 29^{22} + \left(10 a + 12\right)\cdot 29^{23} + \left(19 a + 7\right)\cdot 29^{24} + \left(9 a + 27\right)\cdot 29^{25} + \left(15 a + 25\right)\cdot 29^{26} + \left(6 a + 2\right)\cdot 29^{28} + \left(7 a + 23\right)\cdot 29^{29} + \left(20 a + 5\right)\cdot 29^{30} + \left(24 a + 5\right)\cdot 29^{31} + 22\cdot 29^{32} + \left(23 a + 17\right)\cdot 29^{33} + \left(21 a + 5\right)\cdot 29^{34} + \left(8 a + 3\right)\cdot 29^{35} + \left(11 a + 14\right)\cdot 29^{36} + 27\cdot 29^{37} + \left(2 a + 25\right)\cdot 29^{38} + \left(19 a + 2\right)\cdot 29^{39} + \left(9 a + 5\right)\cdot 29^{40} + \left(15 a + 7\right)\cdot 29^{41} + \left(19 a + 14\right)\cdot 29^{42} + \left(9 a + 9\right)\cdot 29^{43} + \left(8 a + 10\right)\cdot 29^{44} + \left(6 a + 8\right)\cdot 29^{45} + \left(10 a + 25\right)\cdot 29^{46} + \left(18 a + 22\right)\cdot 29^{47} + \left(5 a + 27\right)\cdot 29^{48} + \left(14 a + 22\right)\cdot 29^{49} + \left(24 a + 6\right)\cdot 29^{50} + 29^{51} + \left(21 a + 4\right)\cdot 29^{52} + \left(21 a + 4\right)\cdot 29^{53} + \left(5 a + 11\right)\cdot 29^{54} + \left(18 a + 9\right)\cdot 29^{55} + 8 a\cdot 29^{56} + \left(13 a + 14\right)\cdot 29^{57} + \left(7 a + 4\right)\cdot 29^{58} + \left(28 a + 2\right)\cdot 29^{59} + \left(18 a + 20\right)\cdot 29^{60} + \left(6 a + 27\right)\cdot 29^{61} + \left(8 a + 23\right)\cdot 29^{62} + \left(2 a + 27\right)\cdot 29^{63} + \left(10 a + 1\right)\cdot 29^{64} + \left(20 a + 14\right)\cdot 29^{65} + \left(9 a + 26\right)\cdot 29^{66} + \left(21 a + 20\right)\cdot 29^{67} + \left(13 a + 17\right)\cdot 29^{68} + \left(14 a + 1\right)\cdot 29^{69} + \left(24 a + 12\right)\cdot 29^{70} + \left(5 a + 18\right)\cdot 29^{71} + \left(2 a + 10\right)\cdot 29^{72} + \left(10 a + 11\right)\cdot 29^{73} + \left(24 a + 22\right)\cdot 29^{74} + \left(19 a + 9\right)\cdot 29^{75} + \left(3 a + 18\right)\cdot 29^{76} + \left(21 a + 4\right)\cdot 29^{77} + \left(a + 7\right)\cdot 29^{78} + \left(23 a + 18\right)\cdot 29^{79} + \left(23 a + 16\right)\cdot 29^{80} + \left(5 a + 13\right)\cdot 29^{81} + \left(22 a + 15\right)\cdot 29^{82} + \left(6 a + 15\right)\cdot 29^{83} + \left(5 a + 19\right)\cdot 29^{84} +O\left(29^{ 85 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 a + 13 + \left(22 a + 25\right)\cdot 29 + \left(17 a + 7\right)\cdot 29^{2} + \left(26 a + 15\right)\cdot 29^{3} + \left(27 a + 15\right)\cdot 29^{4} + \left(10 a + 10\right)\cdot 29^{5} + \left(25 a + 12\right)\cdot 29^{6} + \left(19 a + 7\right)\cdot 29^{7} + 17\cdot 29^{8} + \left(25 a + 23\right)\cdot 29^{9} + \left(9 a + 16\right)\cdot 29^{10} + \left(20 a + 10\right)\cdot 29^{11} + \left(7 a + 28\right)\cdot 29^{12} + \left(8 a + 20\right)\cdot 29^{13} + \left(6 a + 1\right)\cdot 29^{14} + \left(19 a + 19\right)\cdot 29^{15} + \left(10 a + 26\right)\cdot 29^{16} + \left(17 a + 5\right)\cdot 29^{17} + \left(a + 13\right)\cdot 29^{18} + \left(23 a + 9\right)\cdot 29^{19} + \left(25 a + 25\right)\cdot 29^{20} + 23\cdot 29^{21} + \left(13 a + 21\right)\cdot 29^{22} + \left(21 a + 12\right)\cdot 29^{23} + \left(20 a + 16\right)\cdot 29^{24} + 28\cdot 29^{26} + \left(10 a + 8\right)\cdot 29^{27} + \left(28 a + 5\right)\cdot 29^{28} + \left(24 a + 10\right)\cdot 29^{29} + \left(10 a + 7\right)\cdot 29^{30} + 26\cdot 29^{31} + \left(2 a + 21\right)\cdot 29^{32} + \left(22 a + 2\right)\cdot 29^{33} + \left(13 a + 23\right)\cdot 29^{34} + \left(7 a + 20\right)\cdot 29^{35} + \left(9 a + 2\right)\cdot 29^{36} + 22 a\cdot 29^{37} + \left(4 a + 26\right)\cdot 29^{38} + \left(20 a + 15\right)\cdot 29^{39} + \left(14 a + 9\right)\cdot 29^{40} + \left(20 a + 16\right)\cdot 29^{41} + \left(12 a + 11\right)\cdot 29^{42} + \left(24 a + 1\right)\cdot 29^{43} + \left(2 a + 14\right)\cdot 29^{44} + \left(4 a + 1\right)\cdot 29^{45} + 28 a\cdot 29^{46} + \left(26 a + 6\right)\cdot 29^{47} + \left(17 a + 27\right)\cdot 29^{48} + \left(14 a + 6\right)\cdot 29^{49} + \left(15 a + 2\right)\cdot 29^{50} + \left(14 a + 27\right)\cdot 29^{51} + \left(8 a + 4\right)\cdot 29^{52} + \left(12 a + 19\right)\cdot 29^{53} + a\cdot 29^{54} + \left(2 a + 16\right)\cdot 29^{55} + \left(21 a + 16\right)\cdot 29^{56} + \left(14 a + 12\right)\cdot 29^{57} + \left(22 a + 20\right)\cdot 29^{58} + \left(11 a + 20\right)\cdot 29^{59} + \left(5 a + 2\right)\cdot 29^{60} + \left(24 a + 27\right)\cdot 29^{61} + \left(16 a + 11\right)\cdot 29^{62} + \left(13 a + 18\right)\cdot 29^{63} + \left(4 a + 23\right)\cdot 29^{64} + \left(10 a + 27\right)\cdot 29^{65} + \left(4 a + 25\right)\cdot 29^{66} + \left(4 a + 21\right)\cdot 29^{67} + \left(20 a + 6\right)\cdot 29^{68} + \left(28 a + 12\right)\cdot 29^{69} + \left(24 a + 25\right)\cdot 29^{70} + \left(15 a + 15\right)\cdot 29^{71} + \left(9 a + 19\right)\cdot 29^{72} + \left(11 a + 8\right)\cdot 29^{73} + \left(10 a + 28\right)\cdot 29^{74} + \left(26 a + 14\right)\cdot 29^{75} + \left(24 a + 10\right)\cdot 29^{76} + \left(23 a + 2\right)\cdot 29^{77} + \left(21 a + 5\right)\cdot 29^{78} + \left(20 a + 2\right)\cdot 29^{79} + \left(15 a + 17\right)\cdot 29^{80} + \left(12 a + 10\right)\cdot 29^{81} + \left(27 a + 1\right)\cdot 29^{82} + \left(11 a + 14\right)\cdot 29^{83} + \left(16 a + 6\right)\cdot 29^{84} +O\left(29^{ 85 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 a + 11 + \left(18 a + 2\right)\cdot 29 + \left(16 a + 15\right)\cdot 29^{2} + \left(11 a + 21\right)\cdot 29^{3} + \left(20 a + 19\right)\cdot 29^{4} + \left(11 a + 21\right)\cdot 29^{5} + \left(23 a + 13\right)\cdot 29^{6} + \left(14 a + 28\right)\cdot 29^{7} + \left(16 a + 12\right)\cdot 29^{8} + \left(15 a + 24\right)\cdot 29^{9} + \left(15 a + 16\right)\cdot 29^{10} + \left(12 a + 10\right)\cdot 29^{11} + \left(a + 9\right)\cdot 29^{12} + \left(23 a + 26\right)\cdot 29^{13} + \left(8 a + 22\right)\cdot 29^{14} + \left(22 a + 19\right)\cdot 29^{15} + \left(24 a + 27\right)\cdot 29^{16} + \left(18 a + 19\right)\cdot 29^{17} + \left(6 a + 19\right)\cdot 29^{18} + 13 a\cdot 29^{19} + \left(8 a + 1\right)\cdot 29^{20} + \left(2 a + 3\right)\cdot 29^{21} + \left(3 a + 23\right)\cdot 29^{22} + \left(10 a + 21\right)\cdot 29^{23} + \left(3 a + 24\right)\cdot 29^{24} + \left(2 a + 13\right)\cdot 29^{25} + \left(24 a + 23\right)\cdot 29^{26} + \left(21 a + 18\right)\cdot 29^{27} + \left(15 a + 4\right)\cdot 29^{28} + \left(23 a + 28\right)\cdot 29^{29} + \left(8 a + 5\right)\cdot 29^{30} + \left(25 a + 6\right)\cdot 29^{31} + \left(15 a + 11\right)\cdot 29^{32} + \left(19 a + 16\right)\cdot 29^{33} + \left(4 a + 13\right)\cdot 29^{34} + \left(18 a + 22\right)\cdot 29^{35} + \left(22 a + 22\right)\cdot 29^{36} + 24\cdot 29^{37} + \left(4 a + 14\right)\cdot 29^{38} + \left(19 a + 3\right)\cdot 29^{39} + \left(a + 3\right)\cdot 29^{40} + 26\cdot 29^{41} + \left(23 a + 5\right)\cdot 29^{42} + \left(21 a + 27\right)\cdot 29^{43} + \left(27 a + 7\right)\cdot 29^{44} + \left(a + 2\right)\cdot 29^{45} + \left(4 a + 12\right)\cdot 29^{46} + \left(13 a + 9\right)\cdot 29^{47} + 22 a\cdot 29^{48} + \left(12 a + 19\right)\cdot 29^{49} + \left(17 a + 18\right)\cdot 29^{50} + \left(28 a + 22\right)\cdot 29^{51} + \left(21 a + 23\right)\cdot 29^{52} + \left(a + 14\right)\cdot 29^{53} + \left(23 a + 17\right)\cdot 29^{54} + \left(17 a + 5\right)\cdot 29^{55} + \left(19 a + 7\right)\cdot 29^{56} + \left(16 a + 22\right)\cdot 29^{57} + \left(10 a + 18\right)\cdot 29^{58} + \left(17 a + 19\right)\cdot 29^{59} + \left(22 a + 14\right)\cdot 29^{60} + \left(18 a + 1\right)\cdot 29^{61} + \left(9 a + 28\right)\cdot 29^{62} + \left(5 a + 25\right)\cdot 29^{63} + \left(24 a + 11\right)\cdot 29^{64} + 13\cdot 29^{65} + \left(14 a + 12\right)\cdot 29^{66} + \left(9 a + 22\right)\cdot 29^{67} + 17\cdot 29^{68} + \left(7 a + 11\right)\cdot 29^{69} + \left(10 a + 27\right)\cdot 29^{70} + \left(28 a + 4\right)\cdot 29^{71} + \left(18 a + 20\right)\cdot 29^{72} + \left(24 a + 18\right)\cdot 29^{73} + \left(2 a + 25\right)\cdot 29^{74} + \left(27 a + 24\right)\cdot 29^{75} + \left(4 a + 28\right)\cdot 29^{76} + \left(27 a + 25\right)\cdot 29^{77} + \left(13 a + 20\right)\cdot 29^{78} + \left(8 a + 24\right)\cdot 29^{79} + \left(27 a + 23\right)\cdot 29^{80} + \left(4 a + 15\right)\cdot 29^{81} + \left(26 a + 5\right)\cdot 29^{82} + \left(7 a + 8\right)\cdot 29^{83} + 27\cdot 29^{84} +O\left(29^{ 85 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 13 + 13\cdot 29 + 29^{2} + 11\cdot 29^{3} + 12\cdot 29^{4} + 24\cdot 29^{5} + 6\cdot 29^{6} + 17\cdot 29^{7} + 18\cdot 29^{9} + 26\cdot 29^{10} + 24\cdot 29^{11} + 26\cdot 29^{12} + 15\cdot 29^{13} + 9\cdot 29^{14} + 26\cdot 29^{15} + 5\cdot 29^{16} + 29^{17} + 22\cdot 29^{18} + 12\cdot 29^{19} + 12\cdot 29^{20} + 8\cdot 29^{21} + 15\cdot 29^{22} + 27\cdot 29^{23} + 22\cdot 29^{24} + 9\cdot 29^{25} + 13\cdot 29^{26} + 13\cdot 29^{27} + 28\cdot 29^{28} + 20\cdot 29^{29} + 7\cdot 29^{30} + 25\cdot 29^{31} + 13\cdot 29^{32} + 21\cdot 29^{33} + 14\cdot 29^{34} + 9\cdot 29^{36} + 29^{37} + 2\cdot 29^{38} + 14\cdot 29^{39} + 6\cdot 29^{40} + 25\cdot 29^{41} + 21\cdot 29^{42} + 20\cdot 29^{43} + 18\cdot 29^{44} + 2\cdot 29^{45} + 15\cdot 29^{46} + 4\cdot 29^{47} + 10\cdot 29^{48} + 3\cdot 29^{49} + 6\cdot 29^{50} + 23\cdot 29^{51} + 17\cdot 29^{52} + 13\cdot 29^{53} + 23\cdot 29^{54} + 4\cdot 29^{55} + 11\cdot 29^{56} + 16\cdot 29^{57} + 5\cdot 29^{58} + 8\cdot 29^{59} + 24\cdot 29^{60} + 3\cdot 29^{61} + 8\cdot 29^{62} + 29^{63} + 2\cdot 29^{64} + 2\cdot 29^{65} + 25\cdot 29^{66} + 4\cdot 29^{67} + 2\cdot 29^{68} + 11\cdot 29^{69} + 23\cdot 29^{70} + 22\cdot 29^{71} + 5\cdot 29^{72} + 28\cdot 29^{73} + 26\cdot 29^{74} + 6\cdot 29^{75} + 27\cdot 29^{76} + 12\cdot 29^{77} + 27\cdot 29^{78} + 8\cdot 29^{79} + 13\cdot 29^{80} + 11\cdot 29^{81} + 15\cdot 29^{82} + 13\cdot 29^{83} + 14\cdot 29^{84} +O\left(29^{ 85 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 5 + 3\cdot 29 + 15\cdot 29^{2} + 21\cdot 29^{3} + 16\cdot 29^{4} + 8\cdot 29^{5} + 6\cdot 29^{6} + 20\cdot 29^{7} + 14\cdot 29^{8} + 7\cdot 29^{9} + 18\cdot 29^{10} + 28\cdot 29^{11} + 23\cdot 29^{12} + 7\cdot 29^{13} + 10\cdot 29^{14} + 24\cdot 29^{15} + 27\cdot 29^{16} + 5\cdot 29^{17} + 15\cdot 29^{18} + 14\cdot 29^{19} + 13\cdot 29^{20} + 10\cdot 29^{21} + 8\cdot 29^{22} + 23\cdot 29^{23} + 23\cdot 29^{24} + 3\cdot 29^{25} + 24\cdot 29^{26} + 9\cdot 29^{27} + 19\cdot 29^{28} + 4\cdot 29^{29} + 13\cdot 29^{30} + 6\cdot 29^{31} + 6\cdot 29^{32} + 6\cdot 29^{33} + 25\cdot 29^{34} + 6\cdot 29^{35} + 20\cdot 29^{36} + 22\cdot 29^{37} + 7\cdot 29^{38} + 8\cdot 29^{39} + 2\cdot 29^{40} + 12\cdot 29^{41} + 22\cdot 29^{42} + 26\cdot 29^{43} + 9\cdot 29^{44} + 8\cdot 29^{45} + 17\cdot 29^{47} + 26\cdot 29^{48} + 26\cdot 29^{49} + 11\cdot 29^{50} + 2\cdot 29^{51} + 22\cdot 29^{52} + 14\cdot 29^{53} + 5\cdot 29^{54} + 5\cdot 29^{55} + 22\cdot 29^{56} + 29^{57} + 10\cdot 29^{58} + 8\cdot 29^{59} + 13\cdot 29^{60} + 29^{61} + 29^{62} + 15\cdot 29^{63} + 11\cdot 29^{64} + 29^{65} + 26\cdot 29^{66} + 7\cdot 29^{67} + 9\cdot 29^{68} + 11\cdot 29^{69} + 4\cdot 29^{70} + 27\cdot 29^{71} + 21\cdot 29^{72} + 13\cdot 29^{73} + 26\cdot 29^{74} + 28\cdot 29^{75} + 23\cdot 29^{76} + 28\cdot 29^{77} + 22\cdot 29^{78} + 24\cdot 29^{79} + 24\cdot 29^{80} + 3\cdot 29^{81} + 19\cdot 29^{82} + 27\cdot 29^{83} + 27\cdot 29^{84} +O\left(29^{ 85 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,4,5)$ |
| $(1,3)$ |
| $(1,2)(3,6)(4,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$12$ |
| $6$ |
$2$ |
$(2,7)(6,8)$ |
$4$ |
| $9$ |
$2$ |
$(1,4)(2,7)(3,5)(6,8)$ |
$-4$ |
| $12$ |
$2$ |
$(1,3)$ |
$-2$ |
| $24$ |
$2$ |
$(1,2)(3,6)(4,7)(5,8)$ |
$0$ |
| $36$ |
$2$ |
$(1,3)(2,6)$ |
$0$ |
| $36$ |
$2$ |
$(1,3)(2,7)(6,8)$ |
$-2$ |
| $16$ |
$3$ |
$(1,4,5)$ |
$-3$ |
| $64$ |
$3$ |
$(1,4,5)(6,7,8)$ |
$0$ |
| $12$ |
$4$ |
$(2,6,7,8)$ |
$2$ |
| $36$ |
$4$ |
$(1,3,4,5)(2,6,7,8)$ |
$0$ |
| $36$ |
$4$ |
$(1,3,4,5)(2,7)(6,8)$ |
$2$ |
| $72$ |
$4$ |
$(1,2,4,7)(3,6,5,8)$ |
$0$ |
| $72$ |
$4$ |
$(1,3)(2,6,7,8)$ |
$0$ |
| $144$ |
$4$ |
$(1,6,3,2)(4,7)(5,8)$ |
$0$ |
| $48$ |
$6$ |
$(1,5,4)(2,7)(6,8)$ |
$1$ |
| $96$ |
$6$ |
$(1,3)(6,8,7)$ |
$1$ |
| $192$ |
$6$ |
$(1,6,4,7,5,8)(2,3)$ |
$0$ |
| $144$ |
$8$ |
$(1,2,3,6,4,7,5,8)$ |
$0$ |
| $96$ |
$12$ |
$(1,4,5)(2,6,7,8)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
