Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 17\cdot 43 + 16\cdot 43^{2} + 2\cdot 43^{3} + 28\cdot 43^{4} + 14\cdot 43^{5} + 32\cdot 43^{6} + 18\cdot 43^{7} + 41\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 36 a + 23 + 17\cdot 43 + \left(42 a + 12\right)\cdot 43^{2} + \left(4 a + 40\right)\cdot 43^{3} + \left(4 a + 26\right)\cdot 43^{4} + \left(38 a + 1\right)\cdot 43^{5} + \left(38 a + 16\right)\cdot 43^{6} + \left(40 a + 23\right)\cdot 43^{7} + \left(3 a + 25\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ a + 17 + \left(7 a + 31\right)\cdot 43 + \left(21 a + 27\right)\cdot 43^{2} + \left(11 a + 3\right)\cdot 43^{3} + \left(13 a + 28\right)\cdot 43^{4} + \left(29 a + 27\right)\cdot 43^{5} + \left(26 a + 6\right)\cdot 43^{6} + \left(29 a + 32\right)\cdot 43^{7} + \left(4 a + 34\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a + 16 + \left(42 a + 25\right)\cdot 43 + 10\cdot 43^{2} + \left(38 a + 3\right)\cdot 43^{3} + \left(38 a + 26\right)\cdot 43^{4} + \left(4 a + 35\right)\cdot 43^{5} + \left(4 a + 16\right)\cdot 43^{6} + \left(2 a + 25\right)\cdot 43^{7} + \left(39 a + 31\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 + 20\cdot 43^{2} + 42\cdot 43^{3} + 32\cdot 43^{4} + 5\cdot 43^{5} + 10\cdot 43^{6} + 37\cdot 43^{7} + 28\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 42 a + 18 + \left(35 a + 37\right)\cdot 43 + \left(21 a + 41\right)\cdot 43^{2} + \left(31 a + 36\right)\cdot 43^{3} + \left(29 a + 29\right)\cdot 43^{4} + 13 a\cdot 43^{5} + \left(16 a + 4\right)\cdot 43^{6} + \left(13 a + 35\right)\cdot 43^{7} + \left(38 a + 9\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(2,4)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,3)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,6,3)(2,4,5)$ | $-1$ |
| $2$ | $6$ | $(1,4,6,5,3,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
