Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{3} + x + 40 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 28 + 27\cdot 43 + 11\cdot 43^{2} + 34\cdot 43^{3} + 8\cdot 43^{4} + 8\cdot 43^{5} + 6\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 a^{2} + 34 a + 17 + \left(11 a^{2} + 16 a + 8\right)\cdot 43 + \left(36 a^{2} + 15 a + 6\right)\cdot 43^{2} + \left(20 a^{2} + 5 a + 15\right)\cdot 43^{3} + \left(2 a^{2} + 37 a + 37\right)\cdot 43^{4} + \left(12 a^{2} + 35 a + 2\right)\cdot 43^{5} + \left(3 a^{2} + 16 a + 32\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a^{2} + 7 a + 3 + \left(2 a^{2} + 15 a + 2\right)\cdot 43 + \left(5 a^{2} + 21 a + 14\right)\cdot 43^{2} + \left(16 a^{2} + 33 a + 26\right)\cdot 43^{3} + \left(20 a^{2} + 9 a + 20\right)\cdot 43^{4} + \left(22 a^{2} + 31 a + 38\right)\cdot 43^{5} + \left(12 a^{2} + 32 a + 23\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 + 36\cdot 43 + 16\cdot 43^{2} + 11\cdot 43^{4} + 7\cdot 43^{5} + 4\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 a^{2} + 27 a + 6 + \left(29 a^{2} + 3 a + 26\right)\cdot 43 + \left(35 a^{2} + 6 a + 3\right)\cdot 43^{2} + \left(9 a^{2} + 8 a + 8\right)\cdot 43^{3} + \left(28 a^{2} + 30 a + 5\right)\cdot 43^{4} + \left(34 a^{2} + 8 a + 23\right)\cdot 43^{5} + \left(21 a^{2} + 38 a + 38\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 12 a^{2} + 2 a + 22 + \left(29 a^{2} + 11 a + 34\right)\cdot 43 + \left(a^{2} + 6 a + 11\right)\cdot 43^{2} + \left(6 a^{2} + 4 a + 5\right)\cdot 43^{3} + \left(20 a^{2} + 39 a + 6\right)\cdot 43^{4} + \left(8 a^{2} + 18 a + 29\right)\cdot 43^{5} + \left(27 a^{2} + 36 a + 33\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 4 a^{2} + 30 a + 15 + \left(5 a^{2} + 2 a + 24\right)\cdot 43 + \left(8 a^{2} + 3 a + 42\right)\cdot 43^{2} + \left(13 a^{2} + 20 a + 38\right)\cdot 43^{3} + \left(33 a^{2} + 40 a + 22\right)\cdot 43^{4} + \left(36 a^{2} + 9 a + 24\right)\cdot 43^{5} + \left(9 a^{2} + 41 a + 30\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 27 a^{2} + 29 a + 16 + \left(8 a^{2} + 36 a + 12\right)\cdot 43 + \left(42 a^{2} + 33 a + 22\right)\cdot 43^{2} + \left(19 a^{2} + 14 a\right)\cdot 43^{3} + \left(24 a^{2} + 15 a + 17\right)\cdot 43^{4} + \left(14 a^{2} + 24 a + 38\right)\cdot 43^{5} + \left(11 a^{2} + 6 a + 2\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,8)(3,5)(6,7)$ |
| $(2,3,6)(5,7,8)$ |
| $(1,6,4,7)(2,3,8,5)$ |
| $(1,2,4,8)(3,6,5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,8)(3,5)(6,7)$ | $-2$ |
| $4$ | $3$ | $(1,3,7)(4,5,6)$ | $\zeta_{3} + 1$ |
| $4$ | $3$ | $(1,7,3)(4,6,5)$ | $-\zeta_{3}$ |
| $6$ | $4$ | $(1,6,4,7)(2,3,8,5)$ | $0$ |
| $4$ | $6$ | $(1,6,3,4,7,5)(2,8)$ | $\zeta_{3}$ |
| $4$ | $6$ | $(1,5,7,4,3,6)(2,8)$ | $-\zeta_{3} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.
