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