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