Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{3} + x + 40 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 a^{2} + 12 a + 36 + \left(26 a^{2} + 33 a + 37\right)\cdot 43 + \left(17 a^{2} + 6 a + 1\right)\cdot 43^{2} + \left(21 a^{2} + 16 a\right)\cdot 43^{3} + \left(38 a^{2} + 31 a + 10\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 a^{2} + 6 a + 37 + \left(13 a^{2} + 30 a + 9\right)\cdot 43 + \left(18 a^{2} + 4 a + 2\right)\cdot 43^{2} + \left(40 a^{2} + 18 a + 23\right)\cdot 43^{3} + \left(40 a^{2} + 25 a + 15\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 a^{2} + 6 a + 38 + \left(17 a^{2} + 22 a + 31\right)\cdot 43 + \left(39 a^{2} + 20 a + 30\right)\cdot 43^{2} + \left(26 a^{2} + 7 a + 3\right)\cdot 43^{3} + \left(24 a^{2} + 40 a + 15\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 a^{2} + 5 a + 11 + \left(33 a^{2} + 15 a + 15\right)\cdot 43 + \left(9 a^{2} + 39 a + 26\right)\cdot 43^{2} + \left(21 a^{2} + 17 a + 3\right)\cdot 43^{3} + \left(39 a^{2} + 6 a + 25\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 a^{2} + 25 a + 2 + \left(42 a^{2} + 30 a + 34\right)\cdot 43 + \left(28 a^{2} + 15 a + 23\right)\cdot 43^{2} + \left(37 a^{2} + 19 a + 39\right)\cdot 43^{3} + \left(22 a^{2} + 14 a + 13\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a^{2} + 34 a + 20 + \left(26 a^{2} + 41 a + 18\right)\cdot 43 + \left(39 a^{2} + 8 a + 16\right)\cdot 43^{2} + \left(22 a^{2} + 26 a + 11\right)\cdot 43^{3} + \left(24 a^{2} + 14 a + 33\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 7 a^{2} + 24 a + 5 + \left(4 a^{2} + 42 a + 10\right)\cdot 43 + \left(24 a^{2} + 17 a + 7\right)\cdot 43^{2} + \left(40 a^{2} + 31 a + 2\right)\cdot 43^{3} + \left(30 a^{2} + 34 a + 5\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 20 a^{2} + 14 a + 28 + \left(5 a^{2} + 28 a + 39\right)\cdot 43 + \left(9 a^{2} + 28 a + 25\right)\cdot 43^{2} + \left(24 a^{2} + 36 a + 5\right)\cdot 43^{3} + \left(15 a^{2} + a + 9\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 31 a^{2} + 3 a + 38 + \left(3 a^{2} + 14 a + 17\right)\cdot 43 + \left(28 a^{2} + 29 a + 37\right)\cdot 43^{2} + \left(22 a^{2} + 41 a + 39\right)\cdot 43^{3} + \left(20 a^{2} + 2 a + 1\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,6)(2,5)(3,9)(4,7)$ |
| $(1,5,3)(2,6,9)(4,7,8)$ |
| $(1,8,6,5,4,9,3,7,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,6)(2,5)(3,9)(4,7)$ | $0$ |
| $2$ | $3$ | $(1,5,3)(2,6,9)(4,7,8)$ | $-1$ |
| $2$ | $9$ | $(1,8,6,5,4,9,3,7,2)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,6,4,3,2,8,5,9,7)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,4,2,5,7,6,3,8,9)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.
