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:
| $r_{ 1 }$ |
$=$ |
$ 41 + 39\cdot 43 + 43^{2} + 16\cdot 43^{3} + 40\cdot 43^{4} + 5\cdot 43^{5} + 17\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 a + 18 + \left(18 a + 21\right)\cdot 43 + \left(35 a + 41\right)\cdot 43^{2} + \left(24 a + 8\right)\cdot 43^{3} + \left(37 a + 32\right)\cdot 43^{4} + \left(36 a + 42\right)\cdot 43^{5} + \left(31 a + 10\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 a + 26 + \left(24 a + 31\right)\cdot 43 + \left(7 a + 15\right)\cdot 43^{2} + \left(18 a + 41\right)\cdot 43^{3} + \left(5 a + 1\right)\cdot 43^{4} + \left(6 a + 42\right)\cdot 43^{5} + \left(11 a + 5\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 a + 27 + \left(17 a + 9\right)\cdot 43 + \left(15 a + 23\right)\cdot 43^{2} + \left(26 a + 19\right)\cdot 43^{3} + \left(41 a + 18\right)\cdot 43^{4} + \left(9 a + 35\right)\cdot 43^{5} + \left(28 a + 16\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 a + 39 + \left(25 a + 14\right)\cdot 43 + \left(27 a + 21\right)\cdot 43^{2} + \left(16 a + 30\right)\cdot 43^{3} + \left(a + 33\right)\cdot 43^{4} + \left(33 a + 3\right)\cdot 43^{5} + \left(14 a + 35\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 + 11\cdot 43 + 25\cdot 43^{2} + 12\cdot 43^{3} + 2\cdot 43^{4} + 42\cdot 43^{5} + 42\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(4,6)$ |
| $(1,6)$ |
| $(1,3,2)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,4)$ |
$-3$ |
| $3$ |
$2$ |
$(1,6)$ |
$1$ |
| $3$ |
$2$ |
$(1,6)(3,4)$ |
$-1$ |
| $6$ |
$2$ |
$(2,3)(4,5)$ |
$-1$ |
| $6$ |
$2$ |
$(1,6)(2,3)(4,5)$ |
$1$ |
| $8$ |
$3$ |
$(1,3,2)(4,5,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,4,6,3)$ |
$-1$ |
| $6$ |
$4$ |
$(1,6)(2,4,5,3)$ |
$1$ |
| $8$ |
$6$ |
$(1,4,5,6,3,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
