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