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