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