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^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 40 a + 3 + \left(28 a + 33\right)\cdot 43 + \left(a + 29\right)\cdot 43^{2} + \left(9 a + 38\right)\cdot 43^{3} + \left(31 a + 23\right)\cdot 43^{4} + \left(15 a + 4\right)\cdot 43^{5} + \left(41 a + 25\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a + \left(14 a + 22\right)\cdot 43 + \left(41 a + 2\right)\cdot 43^{2} + \left(33 a + 3\right)\cdot 43^{3} + \left(11 a + 3\right)\cdot 43^{4} + \left(27 a + 32\right)\cdot 43^{5} + \left(a + 7\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a + 14 + \left(42 a + 39\right)\cdot 43 + \left(19 a + 10\right)\cdot 43^{2} + \left(26 a + 25\right)\cdot 43^{3} + \left(36 a + 37\right)\cdot 43^{4} + \left(32 a + 24\right)\cdot 43^{5} + \left(28 a + 25\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 41 a + 3 + \left(14 a + 38\right)\cdot 43 + \left(2 a + 11\right)\cdot 43^{2} + \left(33 a + 21\right)\cdot 43^{3} + \left(12 a + 18\right)\cdot 43^{4} + \left(42 a + 8\right)\cdot 43^{5} + \left(17 a + 15\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 2 a + 1 + \left(28 a + 12\right)\cdot 43 + \left(40 a + 42\right)\cdot 43^{2} + \left(9 a + 8\right)\cdot 43^{3} + \left(30 a + 41\right)\cdot 43^{4} + 37\cdot 43^{5} + \left(25 a + 33\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 32 a + 25 + 27\cdot 43 + \left(23 a + 31\right)\cdot 43^{2} + \left(16 a + 31\right)\cdot 43^{3} + \left(6 a + 4\right)\cdot 43^{4} + \left(10 a + 21\right)\cdot 43^{5} + \left(14 a + 21\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4,6)(2,3,5)$ |
| $(1,4,6)$ |
| $(1,2,6,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $1$ | $3$ | $(1,6,4)(2,3,5)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,4,6)(2,5,3)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(1,4,6)(2,3,5)$ | $-1$ |
| $2$ | $3$ | $(1,4,6)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,6,4)$ | $\zeta_{3} + 1$ |
| $3$ | $6$ | $(1,2,6,3,4,5)$ | $0$ |
| $3$ | $6$ | $(1,5,4,3,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
