Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 30.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{3} + 2 x + 27 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 a^{2} + 14 a + 10 + \left(22 a^{2} + 22 a + 6\right)\cdot 29 + \left(4 a^{2} + 8 a + 25\right)\cdot 29^{2} + \left(27 a^{2} + 28 a + 12\right)\cdot 29^{3} + \left(4 a^{2} + 2 a\right)\cdot 29^{4} + \left(17 a^{2} + 12 a + 16\right)\cdot 29^{5} + \left(19 a^{2} + 18 a + 15\right)\cdot 29^{6} + \left(2 a^{2} + 11 a + 22\right)\cdot 29^{7} + \left(9 a^{2} + 12 a + 9\right)\cdot 29^{8} + \left(28 a^{2} + 28 a + 27\right)\cdot 29^{9} + \left(22 a^{2} + 6 a + 1\right)\cdot 29^{10} + \left(18 a^{2} + 16 a + 21\right)\cdot 29^{11} + \left(24 a^{2} + 4 a + 4\right)\cdot 29^{12} + \left(5 a^{2} + 5 a + 22\right)\cdot 29^{13} + \left(21 a^{2} + 10 a + 28\right)\cdot 29^{14} + \left(12 a^{2} + 11 a + 11\right)\cdot 29^{15} + \left(23 a^{2} + 2 a + 20\right)\cdot 29^{16} + \left(3 a^{2} + 16 a + 26\right)\cdot 29^{17} + \left(2 a^{2} + 11 a + 11\right)\cdot 29^{18} + \left(7 a^{2} + 26 a + 9\right)\cdot 29^{19} + \left(25 a^{2} + 15 a + 10\right)\cdot 29^{20} + \left(22 a^{2} + 20 a + 15\right)\cdot 29^{21} + \left(25 a^{2} + 13 a + 19\right)\cdot 29^{22} + \left(8 a^{2} + 5 a + 24\right)\cdot 29^{23} + \left(16 a^{2} + 5 a + 23\right)\cdot 29^{24} + \left(3 a^{2} + 26 a + 13\right)\cdot 29^{25} + \left(23 a^{2} + 19 a + 13\right)\cdot 29^{26} + \left(14 a^{2} + 11 a + 3\right)\cdot 29^{27} + \left(21 a^{2} + 27 a + 24\right)\cdot 29^{28} + \left(16 a^{2} + 2 a + 13\right)\cdot 29^{29} +O\left(29^{ 30 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 a^{2} + 6 a + 3 + \left(8 a^{2} + 9 a + 21\right)\cdot 29 + \left(28 a^{2} + 14 a + 23\right)\cdot 29^{2} + \left(18 a^{2} + 20 a + 24\right)\cdot 29^{3} + \left(12 a^{2} + 19 a + 23\right)\cdot 29^{4} + \left(9 a^{2} + 19 a + 8\right)\cdot 29^{5} + \left(16 a^{2} + 19 a + 15\right)\cdot 29^{6} + \left(18 a^{2} + 22 a + 25\right)\cdot 29^{7} + \left(5 a^{2} + 10 a + 26\right)\cdot 29^{8} + \left(19 a^{2} + 16 a + 26\right)\cdot 29^{9} + \left(21 a^{2} + 7 a + 11\right)\cdot 29^{10} + \left(9 a^{2} + 12 a + 15\right)\cdot 29^{11} + \left(6 a^{2} + 20 a + 20\right)\cdot 29^{12} + \left(22 a^{2} + 8 a + 27\right)\cdot 29^{13} + \left(2 a^{2} + 26 a + 28\right)\cdot 29^{14} + \left(24 a^{2} + 28 a + 10\right)\cdot 29^{15} + \left(19 a^{2} + 14 a + 24\right)\cdot 29^{16} + \left(5 a^{2} + 5 a + 21\right)\cdot 29^{17} + \left(25 a^{2} + 18 a + 27\right)\cdot 29^{18} + \left(9 a^{2} + 22 a + 2\right)\cdot 29^{19} + \left(14 a^{2} + 2 a\right)\cdot 29^{20} + \left(5 a^{2} + 12 a + 15\right)\cdot 29^{21} + \left(3 a^{2} + 10 a + 11\right)\cdot 29^{22} + \left(15 a^{2} + 13 a\right)\cdot 29^{23} + \left(14 a^{2} + 7 a + 6\right)\cdot 29^{24} + \left(16 a^{2} + 14 a + 26\right)\cdot 29^{25} + \left(7 a^{2} + 16 a + 25\right)\cdot 29^{26} + \left(7 a^{2} + 12 a + 23\right)\cdot 29^{27} + \left(7 a^{2} + 2 a + 1\right)\cdot 29^{28} + \left(4 a^{2} + 27 a + 21\right)\cdot 29^{29} +O\left(29^{ 30 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a^{2} + 4 a + 28 + \left(11 a^{2} + 22 a + 10\right)\cdot 29 + \left(9 a^{2} + 17 a + 2\right)\cdot 29^{2} + \left(26 a^{2} + 6 a + 2\right)\cdot 29^{3} + \left(13 a^{2} + 17 a + 22\right)\cdot 29^{4} + \left(26 a^{2} + 2 a + 18\right)\cdot 29^{5} + \left(2 a^{2} + 9 a + 12\right)\cdot 29^{6} + \left(15 a^{2} + 20 a\right)\cdot 29^{7} + \left(10 a^{2} + 2\right)\cdot 29^{8} + \left(11 a^{2} + 14 a + 24\right)\cdot 29^{9} + \left(22 a^{2} + 17 a + 10\right)\cdot 29^{10} + \left(22 a + 16\right)\cdot 29^{11} + \left(4 a^{2} + 15 a + 25\right)\cdot 29^{12} + \left(9 a^{2} + 10 a + 16\right)\cdot 29^{13} + \left(7 a^{2} + 16 a\right)\cdot 29^{14} + \left(12 a^{2} + 17 a + 21\right)\cdot 29^{15} + \left(17 a^{2} + 6 a + 2\right)\cdot 29^{16} + \left(17 a^{2} + 12 a + 16\right)\cdot 29^{17} + \left(20 a^{2} + 7\right)\cdot 29^{18} + \left(14 a^{2} + 4 a\right)\cdot 29^{19} + \left(24 a^{2} + a + 19\right)\cdot 29^{20} + \left(17 a^{2} + 19 a + 8\right)\cdot 29^{21} + \left(21 a^{2} + 3 a + 4\right)\cdot 29^{22} + \left(25 a^{2} + 14 a + 18\right)\cdot 29^{23} + \left(19 a^{2} + 12 a + 28\right)\cdot 29^{24} + \left(6 a^{2} + 10 a + 17\right)\cdot 29^{25} + \left(4 a^{2} + 26 a + 7\right)\cdot 29^{26} + \left(20 a^{2} + 23 a + 20\right)\cdot 29^{27} + \left(15 a^{2} + 5 a + 6\right)\cdot 29^{28} + \left(23 a^{2} + 20 a + 13\right)\cdot 29^{29} +O\left(29^{ 30 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 + 25\cdot 29^{2} + 2\cdot 29^{3} + 26\cdot 29^{4} + 27\cdot 29^{5} + 5\cdot 29^{6} + 29^{7} + 9\cdot 29^{8} + 20\cdot 29^{9} + 18\cdot 29^{10} + 25\cdot 29^{11} + 6\cdot 29^{12} + 6\cdot 29^{13} + 9\cdot 29^{14} + 14\cdot 29^{15} + 4\cdot 29^{16} + 13\cdot 29^{17} + 2\cdot 29^{18} + 4\cdot 29^{19} + 29^{21} + 24\cdot 29^{22} + 16\cdot 29^{23} + 13\cdot 29^{24} + 4\cdot 29^{25} + 6\cdot 29^{26} + 22\cdot 29^{27} + 16\cdot 29^{28} + 5\cdot 29^{29} +O\left(29^{ 30 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 a^{2} + 24 a + 16 + \left(21 a^{2} + 18 a\right)\cdot 29 + \left(18 a^{2} + 8 a + 11\right)\cdot 29^{2} + \left(5 a^{2} + 24 a + 26\right)\cdot 29^{3} + \left(15 a^{2} + 7\right)\cdot 29^{4} + \left(7 a^{2} + 20 a + 6\right)\cdot 29^{5} + \left(11 a^{2} + 4 a + 18\right)\cdot 29^{6} + \left(12 a^{2} + 11 a + 7\right)\cdot 29^{7} + \left(25 a^{2} + 11 a + 24\right)\cdot 29^{8} + \left(16 a^{2} + 19 a + 23\right)\cdot 29^{9} + \left(5 a^{2} + 22 a + 9\right)\cdot 29^{10} + \left(15 a^{2} + 7 a + 3\right)\cdot 29^{11} + \left(3 a^{2} + 18 a + 7\right)\cdot 29^{12} + \left(21 a^{2} + 5 a + 26\right)\cdot 29^{13} + \left(21 a + 6\right)\cdot 29^{14} + \left(17 a^{2} + 14 a + 11\right)\cdot 29^{15} + \left(13 a^{2} + 5 a + 6\right)\cdot 29^{16} + \left(26 a^{2} + 22 a + 1\right)\cdot 29^{17} + \left(17 a^{2} + 27 a + 18\right)\cdot 29^{18} + \left(27 a^{2} + 28 a + 26\right)\cdot 29^{19} + \left(2 a^{2} + 7 a + 13\right)\cdot 29^{20} + \left(15 a^{2} + 24 a + 8\right)\cdot 29^{21} + \left(22 a^{2} + 7 a + 8\right)\cdot 29^{22} + \left(15 a^{2} + 15 a + 1\right)\cdot 29^{23} + \left(17 a^{2} + 19 a + 10\right)\cdot 29^{24} + \left(22 a^{2} + 24 a + 5\right)\cdot 29^{25} + \left(11 a^{2} + 25 a + 12\right)\cdot 29^{26} + \left(12 a^{2} + 3 a + 11\right)\cdot 29^{27} + \left(10 a^{2} + 10 a + 25\right)\cdot 29^{28} + \left(13 a^{2} + 25 a + 13\right)\cdot 29^{29} +O\left(29^{ 30 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 18 a^{2} + 11 a + 26 + \left(24 a^{2} + 13 a + 28\right)\cdot 29 + \left(14 a^{2} + 2 a + 28\right)\cdot 29^{2} + \left(4 a^{2} + 23 a + 1\right)\cdot 29^{3} + \left(10 a^{2} + 8 a + 17\right)\cdot 29^{4} + \left(14 a^{2} + 14 a + 2\right)\cdot 29^{5} + \left(6 a^{2} + a + 27\right)\cdot 29^{6} + \left(11 a^{2} + 26 a + 4\right)\cdot 29^{7} + \left(9 a^{2} + 15 a + 10\right)\cdot 29^{8} + \left(18 a^{2} + 15 a + 4\right)\cdot 29^{9} + \left(12 a^{2} + 4 a + 17\right)\cdot 29^{10} + \left(9 a^{2} + 19 a + 8\right)\cdot 29^{11} + \left(8 a + 1\right)\cdot 29^{12} + \left(14 a^{2} + 13 a + 4\right)\cdot 29^{13} + \left(2 a + 1\right)\cdot 29^{14} + \left(4 a^{2} + 10\right)\cdot 29^{15} + \left(17 a^{2} + 20 a + 2\right)\cdot 29^{16} + \left(7 a^{2} + 22\right)\cdot 29^{17} + \left(6 a^{2} + 17 a + 7\right)\cdot 29^{18} + \left(7 a^{2} + 27 a + 19\right)\cdot 29^{19} + \left(8 a^{2} + 11 a + 16\right)\cdot 29^{20} + \left(17 a^{2} + 18 a + 17\right)\cdot 29^{21} + \left(10 a^{2} + 11 a + 18\right)\cdot 29^{22} + \left(23 a^{2} + 9 a + 24\right)\cdot 29^{23} + \left(21 a^{2} + 11 a + 11\right)\cdot 29^{24} + \left(18 a^{2} + 21 a + 24\right)\cdot 29^{25} + \left(a^{2} + 11 a + 13\right)\cdot 29^{26} + \left(23 a^{2} + 22 a + 14\right)\cdot 29^{27} + \left(20 a^{2} + 24 a + 13\right)\cdot 29^{28} + \left(17 a^{2} + 5 a + 5\right)\cdot 29^{29} +O\left(29^{ 30 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 21 a^{2} + 28 a + 8 + \left(27 a^{2} + 8\right)\cdot 29 + \left(10 a^{2} + 6 a + 10\right)\cdot 29^{2} + \left(4 a^{2} + 13 a + 5\right)\cdot 29^{3} + \left(a^{2} + 8 a + 18\right)\cdot 29^{4} + \left(12 a^{2} + 18 a + 2\right)\cdot 29^{5} + \left(a^{2} + 4 a + 5\right)\cdot 29^{6} + \left(27 a^{2} + 24 a + 27\right)\cdot 29^{7} + \left(26 a^{2} + 6 a + 6\right)\cdot 29^{8} + \left(21 a^{2} + 22 a + 11\right)\cdot 29^{9} + \left(a^{2} + 27 a + 14\right)\cdot 29^{10} + \left(4 a^{2} + 8 a + 17\right)\cdot 29^{11} + \left(19 a^{2} + 19 a + 8\right)\cdot 29^{12} + \left(14 a^{2} + 14 a + 27\right)\cdot 29^{13} + \left(25 a^{2} + 10 a + 10\right)\cdot 29^{14} + \left(16 a^{2} + 14 a + 1\right)\cdot 29^{15} + \left(24 a^{2} + 8 a + 21\right)\cdot 29^{16} + \left(25 a^{2} + a + 19\right)\cdot 29^{17} + \left(14 a^{2} + 12 a + 23\right)\cdot 29^{18} + \left(20 a^{2} + 6 a + 26\right)\cdot 29^{19} + \left(11 a^{2} + 18 a + 15\right)\cdot 29^{20} + \left(8 a^{2} + 21 a + 28\right)\cdot 29^{21} + \left(3 a^{2} + 10 a + 1\right)\cdot 29^{22} + \left(27 a^{2} + 26\right)\cdot 29^{23} + \left(25 a^{2} + 2 a + 1\right)\cdot 29^{24} + \left(18 a^{2} + 19 a + 10\right)\cdot 29^{25} + \left(9 a^{2} + 15 a + 9\right)\cdot 29^{26} + \left(9 a^{2} + 12 a + 7\right)\cdot 29^{27} + \left(11 a^{2} + 16 a + 7\right)\cdot 29^{28} + \left(11 a^{2} + 5 a + 11\right)\cdot 29^{29} +O\left(29^{ 30 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 13 + 10\cdot 29 + 18\cdot 29^{2} + 10\cdot 29^{3} + 4\cdot 29^{5} + 16\cdot 29^{6} + 26\cdot 29^{7} + 26\cdot 29^{8} + 6\cdot 29^{9} + 2\cdot 29^{10} + 8\cdot 29^{11} + 12\cdot 29^{12} + 14\cdot 29^{13} + 6\cdot 29^{15} + 5\cdot 29^{16} + 24\cdot 29^{17} + 16\cdot 29^{18} + 26\cdot 29^{19} + 10\cdot 29^{20} + 21\cdot 29^{21} + 27\cdot 29^{22} + 3\cdot 29^{23} + 20\cdot 29^{24} + 13\cdot 29^{25} + 27\cdot 29^{26} + 12\cdot 29^{27} + 20\cdot 29^{28} + 2\cdot 29^{29} +O\left(29^{ 30 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(6,8)$ |
| $(1,8,6)(4,5,7)$ |
| $(1,5)(2,8)(3,4)(6,7)$ |
| $(2,7)(4,5)$ |
| $(2,4)(5,7)$ |
| $(1,8)(3,6)$ |
| $(1,6)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $3$ |
$2$ |
$(1,8)(2,5)(3,6)(4,7)$ |
$-2$ |
| $4$ |
$2$ |
$(1,5)(2,8)(3,4)(6,7)$ |
$0$ |
| $6$ |
$2$ |
$(2,4)(5,7)$ |
$2$ |
| $6$ |
$2$ |
$(1,8)(2,4)(3,6)(5,7)$ |
$-2$ |
| $12$ |
$2$ |
$(1,6)(5,7)$ |
$-2$ |
| $12$ |
$2$ |
$(1,7)(2,8)(3,4)(5,6)$ |
$0$ |
| $32$ |
$3$ |
$(2,4,5)(3,8,6)$ |
$0$ |
| $12$ |
$4$ |
$(1,5,8,2)(3,4,6,7)$ |
$0$ |
| $12$ |
$4$ |
$(1,4,3,5)(2,6,7,8)$ |
$0$ |
| $12$ |
$4$ |
$(1,3,6,8)(2,7,4,5)$ |
$2$ |
| $24$ |
$4$ |
$(1,6)(2,7,4,5)$ |
$0$ |
| $24$ |
$4$ |
$(1,7,8,2)(3,4,6,5)$ |
$0$ |
| $32$ |
$6$ |
$(1,7)(2,6,4,3,5,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
