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:
| $r_{ 1 }$ |
$=$ |
$ 29 a^{2} + 4 a + 38 + \left(3 a^{2} + 38 a + 25\right)\cdot 43 + \left(3 a^{2} + 25 a + 10\right)\cdot 43^{2} + \left(19 a^{2} + 37 a\right)\cdot 43^{3} + \left(31 a^{2} + 22 a + 29\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 29 a^{2} + 24 a + 18 + \left(7 a^{2} + 40 a + 5\right)\cdot 43 + \left(19 a^{2} + 33 a + 2\right)\cdot 43^{2} + \left(27 a^{2} + 10 a + 2\right)\cdot 43^{3} + \left(42 a^{2} + 32 a + 13\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 a^{2} + 8 a + 24 + \left(11 a^{2} + 12\right)\cdot 43 + \left(10 a^{2} + 40 a + 23\right)\cdot 43^{2} + \left(24 a^{2} + 5 a + 30\right)\cdot 43^{3} + \left(22 a^{2} + a + 36\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a^{2} + 14 + \left(8 a^{2} + 41 a + 10\right)\cdot 43 + \left(26 a^{2} + 21 a + 5\right)\cdot 43^{2} + \left(30 a^{2} + 8 a + 6\right)\cdot 43^{3} + \left(34 a^{2} + 34 a + 16\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ a^{2} + 13 a + 28 + \left(26 a^{2} + 23 a + 17\right)\cdot 43 + \left(19 a^{2} + 5 a + 2\right)\cdot 43^{2} + \left(23 a^{2} + 22 a + 28\right)\cdot 43^{3} + \left(26 a^{2} + 16\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 a^{2} + 35 a + 19 + \left(22 a^{2} + 17 a + 38\right)\cdot 43 + \left(31 a^{2} + 31 a\right)\cdot 43^{2} + \left(6 a^{2} + 22 a + 35\right)\cdot 43^{3} + \left(27 a^{2} + 17 a + 11\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 35 a^{2} + 4 a + 42 + \left(16 a^{2} + 30 a + 5\right)\cdot 43 + \left(8 a^{2} + 28 a + 14\right)\cdot 43^{2} + \left(17 a^{2} + 25 a + 13\right)\cdot 43^{3} + \left(27 a^{2} + 2 a + 26\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 13 a^{2} + 6 a + 36 + \left(9 a^{2} + 22 a + 20\right)\cdot 43 + \left(4 a^{2} + 3 a + 6\right)\cdot 43^{2} + \left(35 a^{2} + 10 a + 7\right)\cdot 43^{3} + \left(16 a^{2} + 10 a + 10\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 20 a^{2} + 35 a + 39 + \left(23 a^{2} + a + 34\right)\cdot 43 + \left(6 a^{2} + 24 a + 20\right)\cdot 43^{2} + \left(31 a^{2} + 28 a + 6\right)\cdot 43^{3} + \left(28 a^{2} + 7 a + 12\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,9,2,6,4,8,7,3,5)$ |
| $(1,6,7)(2,8,5)(3,9,4)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
$2$ |
| $9$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
$0$ |
$0$ |
| $2$ |
$3$ |
$(1,6,7)(2,8,5)(3,9,4)$ |
$-1$ |
$-1$ |
$-1$ |
| $2$ |
$9$ |
$(1,9,2,6,4,8,7,3,5)$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ |
$9$ |
$(1,2,4,7,5,9,6,8,3)$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ |
$9$ |
$(1,4,5,6,3,2,7,9,8)$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
