Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 23.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{3} + 2 x + 27 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ a^{2} + 17 + \left(11 a^{2} + 10 a + 5\right)\cdot 29 + \left(9 a^{2} + 22 a + 5\right)\cdot 29^{2} + \left(24 a^{2} + 18 a + 22\right)\cdot 29^{3} + \left(8 a^{2} + 24 a + 21\right)\cdot 29^{4} + \left(17 a^{2} + 18 a + 10\right)\cdot 29^{5} + \left(24 a^{2} + a + 2\right)\cdot 29^{6} + \left(10 a^{2} + 22 a + 21\right)\cdot 29^{7} + \left(25 a^{2} + 3 a + 13\right)\cdot 29^{8} + \left(2 a^{2} + 10 a + 20\right)\cdot 29^{9} + \left(14 a^{2} + a + 2\right)\cdot 29^{10} + \left(5 a^{2} + 2 a + 25\right)\cdot 29^{11} + \left(5 a^{2} + 16 a + 11\right)\cdot 29^{12} + \left(21 a^{2} + 17 a + 15\right)\cdot 29^{13} + \left(13 a^{2} + 27 a + 28\right)\cdot 29^{14} + \left(2 a^{2} + 12 a + 26\right)\cdot 29^{15} + \left(3 a^{2} + 13 a + 11\right)\cdot 29^{16} + \left(25 a^{2} + 26 a + 28\right)\cdot 29^{17} + \left(11 a^{2} + 4 a + 5\right)\cdot 29^{18} + \left(a^{2} + 23 a + 4\right)\cdot 29^{19} + \left(10 a^{2} + 17 a + 14\right)\cdot 29^{20} + \left(4 a^{2} + 9 a + 1\right)\cdot 29^{21} + \left(11 a^{2} + 10 a + 25\right)\cdot 29^{22} +O\left(29^{ 23 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 a^{2} + 26 a + 24 + \left(16 a^{2} + 24 a + 3\right)\cdot 29 + \left(18 a^{2} + 16 a + 27\right)\cdot 29^{2} + \left(24 a^{2} + 4 a + 12\right)\cdot 29^{3} + \left(23 a^{2} + 11 a + 22\right)\cdot 29^{4} + \left(3 a^{2} + 21\right)\cdot 29^{5} + \left(25 a^{2} + 3 a + 12\right)\cdot 29^{6} + \left(22 a^{2} + 18 a + 27\right)\cdot 29^{7} + \left(18 a^{2} + 8 a + 4\right)\cdot 29^{8} + \left(13 a^{2} + 7 a + 25\right)\cdot 29^{9} + \left(22 a + 3\right)\cdot 29^{10} + \left(28 a^{2} + 9 a + 26\right)\cdot 29^{11} + \left(12 a^{2} + 7 a + 2\right)\cdot 29^{12} + \left(2 a^{2} + 16 a\right)\cdot 29^{13} + \left(18 a^{2} + 22 a + 15\right)\cdot 29^{14} + \left(15 a + 24\right)\cdot 29^{15} + \left(22 a^{2} + 26 a + 17\right)\cdot 29^{16} + \left(26 a^{2} + 16 a + 1\right)\cdot 29^{17} + \left(11 a^{2} + a + 6\right)\cdot 29^{18} + \left(13 a^{2} + 2 a + 20\right)\cdot 29^{19} + \left(27 a^{2} + 19 a + 27\right)\cdot 29^{20} + \left(13 a^{2} + 28 a + 23\right)\cdot 29^{21} + \left(19 a^{2} + 12 a + 16\right)\cdot 29^{22} +O\left(29^{ 23 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 a^{2} + 20 a + 23 + \left(3 a^{2} + 4 a + 12\right)\cdot 29 + \left(5 a^{2} + 8 a + 14\right)\cdot 29^{2} + \left(a^{2} + 6 a + 27\right)\cdot 29^{3} + \left(9 a^{2} + 19 a + 28\right)\cdot 29^{4} + \left(25 a^{2} + 3 a + 2\right)\cdot 29^{5} + \left(7 a^{2} + 27 a + 5\right)\cdot 29^{6} + \left(27 a^{2} + 14 a + 5\right)\cdot 29^{7} + \left(11 a^{2} + 10 a + 11\right)\cdot 29^{8} + \left(14 a^{2} + 21 a + 21\right)\cdot 29^{9} + \left(15 a^{2} + 26 a + 28\right)\cdot 29^{10} + \left(3 a^{2} + 24 a + 22\right)\cdot 29^{11} + \left(6 a^{2} + 25 a + 23\right)\cdot 29^{12} + \left(19 a^{2} + 8 a + 18\right)\cdot 29^{13} + \left(9 a^{2} + 9 a + 3\right)\cdot 29^{14} + \left(20 a^{2} + 6 a + 4\right)\cdot 29^{15} + \left(25 a^{2} + 2 a + 11\right)\cdot 29^{16} + \left(20 a^{2} + 20 a + 3\right)\cdot 29^{17} + \left(23 a^{2} + 14 a + 20\right)\cdot 29^{18} + \left(5 a^{2} + 26 a + 6\right)\cdot 29^{19} + \left(21 a^{2} + 28 a + 12\right)\cdot 29^{20} + \left(a^{2} + 27 a + 14\right)\cdot 29^{21} + \left(14 a^{2} + 5 a + 16\right)\cdot 29^{22} +O\left(29^{ 23 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 + 23\cdot 29 + 2\cdot 29^{2} + 16\cdot 29^{3} + 9\cdot 29^{4} + 16\cdot 29^{5} + 18\cdot 29^{6} + 29^{7} + 4\cdot 29^{8} + 13\cdot 29^{9} + 5\cdot 29^{10} + 22\cdot 29^{11} + 27\cdot 29^{12} + 29^{13} + 26\cdot 29^{14} + 19\cdot 29^{15} + 17\cdot 29^{16} + 21\cdot 29^{17} + 3\cdot 29^{18} + 27\cdot 29^{19} + 5\cdot 29^{20} + 25\cdot 29^{21} + 17\cdot 29^{22} +O\left(29^{ 23 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 3 a + 6 + \left(a^{2} + 23 a + 21\right)\cdot 29 + \left(a^{2} + 18 a + 3\right)\cdot 29^{2} + \left(9 a^{2} + 5 a + 21\right)\cdot 29^{3} + \left(25 a^{2} + 22 a + 14\right)\cdot 29^{4} + \left(7 a^{2} + 9 a + 17\right)\cdot 29^{5} + \left(8 a^{2} + 24 a + 9\right)\cdot 29^{6} + \left(24 a^{2} + 17 a\right)\cdot 29^{7} + \left(13 a^{2} + 16 a + 8\right)\cdot 29^{8} + \left(12 a^{2} + 11 a + 4\right)\cdot 29^{9} + \left(14 a^{2} + 5 a + 3\right)\cdot 29^{10} + \left(24 a^{2} + 17 a + 2\right)\cdot 29^{11} + \left(10 a^{2} + 5 a\right)\cdot 29^{12} + \left(5 a^{2} + 24 a + 4\right)\cdot 29^{13} + \left(26 a^{2} + 7 a + 16\right)\cdot 29^{14} + \left(25 a^{2} + 19\right)\cdot 29^{15} + \left(3 a^{2} + 18 a + 22\right)\cdot 29^{16} + \left(6 a^{2} + 14 a + 12\right)\cdot 29^{17} + \left(5 a^{2} + 22 a + 16\right)\cdot 29^{18} + \left(14 a^{2} + 3 a + 11\right)\cdot 29^{19} + \left(20 a^{2} + 21 a + 18\right)\cdot 29^{20} + \left(10 a^{2} + 19 a + 19\right)\cdot 29^{21} + \left(27 a^{2} + 5 a + 17\right)\cdot 29^{22} +O\left(29^{ 23 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 + 22\cdot 29 + 22\cdot 29^{2} + 8\cdot 29^{3} + 25\cdot 29^{4} + 6\cdot 29^{5} + 26\cdot 29^{6} + 5\cdot 29^{7} + 29^{8} + 25\cdot 29^{9} + 19\cdot 29^{10} + 13\cdot 29^{11} + 29^{12} + 23\cdot 29^{13} + 15\cdot 29^{14} + 26\cdot 29^{15} + 2\cdot 29^{16} + 11\cdot 29^{17} + 26\cdot 29^{18} + 4\cdot 29^{19} + 8\cdot 29^{20} + 6\cdot 29^{21} + 26\cdot 29^{22} +O\left(29^{ 23 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 3 a^{2} + 17 a + 17 + \left(10 a^{2} + 16 a + 11\right)\cdot 29 + \left(19 a^{2} + 6 a + 4\right)\cdot 29^{2} + \left(8 a^{2} + 3 a + 18\right)\cdot 29^{3} + \left(20 a^{2} + 23 a + 24\right)\cdot 29^{4} + \left(26 a^{2} + 19 a + 4\right)\cdot 29^{5} + \left(27 a^{2} + 16 a + 22\right)\cdot 29^{6} + \left(27 a^{2} + 8 a + 15\right)\cdot 29^{7} + \left(5 a^{2} + 22\right)\cdot 29^{8} + \left(18 a^{2} + 20 a + 16\right)\cdot 29^{9} + \left(27 a^{2} + 14 a + 25\right)\cdot 29^{10} + \left(4 a^{2} + 25 a + 24\right)\cdot 29^{11} + \left(11 a^{2} + 10 a + 20\right)\cdot 29^{12} + \left(27 a^{2} + 26 a\right)\cdot 29^{13} + \left(25 a^{2} + 20 a + 6\right)\cdot 29^{14} + \left(25 a^{2} + 19 a + 21\right)\cdot 29^{15} + \left(25 a^{2} + 27 a + 1\right)\cdot 29^{16} + \left(4 a^{2} + 24 a + 11\right)\cdot 29^{17} + \left(28 a^{2} + 27 a + 16\right)\cdot 29^{18} + \left(10 a^{2} + 10 a + 13\right)\cdot 29^{19} + \left(22 a^{2} + 7 a + 23\right)\cdot 29^{20} + \left(13 a^{2} + 6 a + 20\right)\cdot 29^{21} + \left(9 a^{2} + 7 a\right)\cdot 29^{22} +O\left(29^{ 23 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 10 + 16\cdot 29 + 2\cdot 29^{2} + 15\cdot 29^{3} + 19\cdot 29^{5} + 4\cdot 29^{6} + 8\cdot 29^{7} + 11\cdot 29^{8} + 21\cdot 29^{9} + 27\cdot 29^{10} + 25\cdot 29^{12} + 4\cdot 29^{13} + 13\cdot 29^{14} + 9\cdot 29^{15} + 25\cdot 29^{16} + 26\cdot 29^{17} + 4\cdot 29^{18} + 22\cdot 29^{19} + 2\cdot 29^{20} + 3\cdot 29^{21} + 19\cdot 29^{22} +O\left(29^{ 23 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 4 a^{2} + 21 a + 28 + \left(15 a^{2} + 7 a + 27\right)\cdot 29 + \left(4 a^{2} + 14 a + 3\right)\cdot 29^{2} + \left(19 a^{2} + 19 a + 3\right)\cdot 29^{3} + \left(28 a^{2} + 15 a + 26\right)\cdot 29^{4} + \left(5 a^{2} + 5 a + 15\right)\cdot 29^{5} + \left(22 a^{2} + 14 a + 14\right)\cdot 29^{6} + \left(2 a^{2} + 5 a + 1\right)\cdot 29^{7} + \left(11 a^{2} + 18 a + 10\right)\cdot 29^{8} + \left(25 a^{2} + 16 a + 26\right)\cdot 29^{9} + \left(14 a^{2} + 16 a + 27\right)\cdot 29^{10} + \left(20 a^{2} + 7 a + 6\right)\cdot 29^{11} + \left(11 a^{2} + 21 a + 2\right)\cdot 29^{12} + \left(11 a^{2} + 22 a + 18\right)\cdot 29^{13} + \left(22 a^{2} + 27 a + 20\right)\cdot 29^{14} + \left(11 a^{2} + 2 a + 21\right)\cdot 29^{15} + \left(6 a^{2} + 28 a + 4\right)\cdot 29^{16} + \left(3 a^{2} + 12 a + 28\right)\cdot 29^{17} + \left(6 a^{2} + 15 a + 15\right)\cdot 29^{18} + \left(12 a^{2} + 20 a + 5\right)\cdot 29^{19} + \left(14 a^{2} + 21 a + 3\right)\cdot 29^{20} + \left(13 a^{2} + 23 a + 1\right)\cdot 29^{21} + \left(5 a^{2} + 15 a + 5\right)\cdot 29^{22} +O\left(29^{ 23 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,5,2)(3,9,7)$ |
| $(1,8)(2,6)(3,7,9)(4,5)$ |
| $(3,9,7)(4,6,8)$ |
| $(3,8,9,4,7,6)$ |
| $(3,7,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $9$ |
$2$ |
$(1,8)(2,6)(4,5)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,2,5)(3,7,9)(4,8,6)$ |
$3 \zeta_{3}$ |
$-3 \zeta_{3} - 3$ |
| $1$ |
$3$ |
$(1,5,2)(3,9,7)(4,6,8)$ |
$-3 \zeta_{3} - 3$ |
$3 \zeta_{3}$ |
| $3$ |
$3$ |
$(1,5,2)(3,9,7)$ |
$\zeta_{3} - 1$ |
$-\zeta_{3} - 2$ |
| $3$ |
$3$ |
$(1,2,5)(3,7,9)$ |
$-\zeta_{3} - 2$ |
$\zeta_{3} - 1$ |
| $3$ |
$3$ |
$(3,9,7)$ |
$2 \zeta_{3} + 1$ |
$-2 \zeta_{3} - 1$ |
| $3$ |
$3$ |
$(3,7,9)$ |
$-2 \zeta_{3} - 1$ |
$2 \zeta_{3} + 1$ |
| $3$ |
$3$ |
$(1,5,2)(3,7,9)(4,6,8)$ |
$-\zeta_{3} + 1$ |
$\zeta_{3} + 2$ |
| $3$ |
$3$ |
$(1,2,5)(3,9,7)(4,8,6)$ |
$\zeta_{3} + 2$ |
$-\zeta_{3} + 1$ |
| $6$ |
$3$ |
$(1,2,5)(4,6,8)$ |
$0$ |
$0$ |
| $18$ |
$3$ |
$(1,9,8)(2,3,6)(4,5,7)$ |
$0$ |
$0$ |
| $9$ |
$6$ |
$(1,8)(2,6)(3,7,9)(4,5)$ |
$-1$ |
$-1$ |
| $9$ |
$6$ |
$(1,8)(2,6)(3,9,7)(4,5)$ |
$-1$ |
$-1$ |
| $9$ |
$6$ |
$(1,4,5,6,2,8)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $9$ |
$6$ |
$(1,8,2,6,5,4)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $9$ |
$6$ |
$(1,6,2,4,5,8)(3,9,7)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $9$ |
$6$ |
$(1,8,5,4,2,6)(3,7,9)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $9$ |
$6$ |
$(1,5,2)(3,4,9,6,7,8)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $9$ |
$6$ |
$(1,2,5)(3,8,7,6,9,4)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $18$ |
$9$ |
$(1,3,4,5,9,6,2,7,8)$ |
$0$ |
$0$ |
| $18$ |
$9$ |
$(1,4,9,2,8,3,5,6,7)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
