Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 20 a + 18 + \left(16 a + 30\right)\cdot 43 + \left(2 a + 41\right)\cdot 43^{2} + \left(22 a + 38\right)\cdot 43^{3} + \left(5 a + 1\right)\cdot 43^{4} + 16\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 a + 13 + \left(32 a + 3\right)\cdot 43 + \left(7 a + 15\right)\cdot 43^{2} + \left(36 a + 22\right)\cdot 43^{3} + \left(41 a + 28\right)\cdot 43^{4} + \left(30 a + 39\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 a + 38 + \left(26 a + 26\right)\cdot 43 + \left(40 a + 27\right)\cdot 43^{2} + \left(20 a + 15\right)\cdot 43^{3} + \left(37 a + 28\right)\cdot 43^{4} + \left(42 a + 10\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 a + 37 + \left(10 a + 11\right)\cdot 43 + \left(35 a + 33\right)\cdot 43^{2} + \left(6 a + 7\right)\cdot 43^{3} + \left(a + 34\right)\cdot 43^{4} + \left(12 a + 28\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 2 + 28\cdot 43 + 18\cdot 43^{2} + 29\cdot 43^{3} + 42\cdot 43^{4} + 22\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 + 28\cdot 43 + 35\cdot 43^{2} + 14\cdot 43^{3} + 36\cdot 43^{4} + 10\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,2)(3,4,6)$ |
| $(1,2)(3,4)$ |
| $(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $3$ | $2$ | $(2,3)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $1$ |
| $8$ | $3$ | $(1,5,2)(3,4,6)$ | $0$ |
| $6$ | $4$ | $(1,3,4,2)$ | $-1$ |
| $6$ | $4$ | $(1,4)(2,6,3,5)$ | $1$ |
| $8$ | $6$ | $(1,5,2,4,6,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
