Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 38 a + 21 + \left(28 a + 6\right)\cdot 43 + \left(17 a + 36\right)\cdot 43^{2} + \left(3 a + 3\right)\cdot 43^{3} + \left(32 a + 41\right)\cdot 43^{4} + \left(28 a + 41\right)\cdot 43^{5} + \left(11 a + 13\right)\cdot 43^{6} + \left(5 a + 10\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 a + 31 + \left(36 a + 41\right)\cdot 43 + \left(25 a + 27\right)\cdot 43^{2} + \left(24 a + 23\right)\cdot 43^{3} + \left(a + 39\right)\cdot 43^{4} + \left(7 a + 21\right)\cdot 43^{5} + \left(35 a + 18\right)\cdot 43^{6} + 11 a\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 a + 21 + \left(14 a + 12\right)\cdot 43 + \left(38 a + 42\right)\cdot 43^{2} + \left(25 a + 7\right)\cdot 43^{3} + \left(33 a + 20\right)\cdot 43^{4} + \left(2 a + 36\right)\cdot 43^{5} + \left(27 a + 35\right)\cdot 43^{6} + \left(23 a + 5\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 a + 16 + \left(14 a + 40\right)\cdot 43 + \left(25 a + 24\right)\cdot 43^{2} + \left(39 a + 32\right)\cdot 43^{3} + \left(10 a + 26\right)\cdot 43^{4} + \left(14 a + 38\right)\cdot 43^{5} + \left(31 a + 39\right)\cdot 43^{6} + \left(37 a + 3\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 a + 7 + \left(6 a + 16\right)\cdot 43 + \left(17 a + 17\right)\cdot 43^{2} + \left(18 a + 22\right)\cdot 43^{3} + \left(41 a + 16\right)\cdot 43^{4} + \left(35 a + 27\right)\cdot 43^{5} + \left(7 a + 3\right)\cdot 43^{6} + \left(31 a + 20\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 a + 36 + \left(28 a + 11\right)\cdot 43 + \left(4 a + 23\right)\cdot 43^{2} + \left(17 a + 38\right)\cdot 43^{3} + \left(9 a + 27\right)\cdot 43^{4} + \left(40 a + 5\right)\cdot 43^{5} + \left(15 a + 17\right)\cdot 43^{6} + \left(19 a + 2\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)(2,3)(4,5)$ |
| $(1,5,3)$ |
| $(2,6,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,6)(2,3)(4,5)$ | $0$ |
| $1$ | $3$ | $(1,5,3)(2,6,4)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,3,5)(2,4,6)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(1,5,3)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,3,5)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,3,5)(2,6,4)$ | $-1$ |
| $3$ | $6$ | $(1,4,5,2,3,6)$ | $0$ |
| $3$ | $6$ | $(1,6,3,2,5,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
