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:
| $r_{ 1 }$ |
$=$ |
$ 10 + 7\cdot 43 + 28\cdot 43^{2} + 5\cdot 43^{3} + 33\cdot 43^{4} + 38\cdot 43^{5} + 12\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 a + 23 + \left(2 a + 8\right)\cdot 43 + \left(35 a + 3\right)\cdot 43^{2} + \left(28 a + 24\right)\cdot 43^{3} + \left(41 a + 8\right)\cdot 43^{4} + \left(3 a + 2\right)\cdot 43^{5} + 15 a\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 a + \left(40 a + 34\right)\cdot 43 + \left(7 a + 35\right)\cdot 43^{2} + \left(14 a + 17\right)\cdot 43^{3} + \left(a + 21\right)\cdot 43^{4} + \left(39 a + 7\right)\cdot 43^{5} + \left(27 a + 11\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 a + 15 + \left(16 a + 7\right)\cdot 43 + \left(12 a + 13\right)\cdot 43^{2} + \left(3 a + 33\right)\cdot 43^{3} + \left(26 a + 37\right)\cdot 43^{4} + \left(4 a + 30\right)\cdot 43^{5} + \left(23 a + 16\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 a + \left(26 a + 39\right)\cdot 43 + \left(30 a + 8\right)\cdot 43^{2} + \left(39 a + 24\right)\cdot 43^{3} + \left(16 a + 17\right)\cdot 43^{4} + \left(38 a + 9\right)\cdot 43^{5} + \left(19 a + 35\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 + 32\cdot 43 + 39\cdot 43^{2} + 23\cdot 43^{3} + 10\cdot 43^{4} + 40\cdot 43^{5} + 9\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)$ |
| $(1,2)(5,6)$ |
| $(1,2,3)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(3,4)$ | $1$ |
| $3$ | $2$ | $(1,6)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,2)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,3,6,4)$ | $-1$ |
| $6$ | $4$ | $(1,6)(2,3,5,4)$ | $1$ |
| $8$ | $6$ | $(1,2,3,6,5,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
