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