Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 13.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 28 + 16\cdot 43 + 32\cdot 43^{2} + 32\cdot 43^{3} + 25\cdot 43^{4} + 16\cdot 43^{5} + 11\cdot 43^{6} + 34\cdot 43^{7} + 19\cdot 43^{8} + 28\cdot 43^{9} + 16\cdot 43^{10} + 30\cdot 43^{11} + 33\cdot 43^{12} +O\left(43^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 33 a + 37 + \left(36 a + 16\right)\cdot 43 + \left(42 a + 32\right)\cdot 43^{2} + 33 a\cdot 43^{3} + \left(13 a + 10\right)\cdot 43^{4} + \left(28 a + 19\right)\cdot 43^{5} + \left(40 a + 9\right)\cdot 43^{6} + \left(39 a + 26\right)\cdot 43^{7} + \left(8 a + 10\right)\cdot 43^{8} + \left(41 a + 1\right)\cdot 43^{9} + \left(17 a + 14\right)\cdot 43^{10} + \left(2 a + 14\right)\cdot 43^{11} + \left(24 a + 33\right)\cdot 43^{12} +O\left(43^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 a + 16 + \left(36 a + 22\right)\cdot 43 + \left(42 a + 4\right)\cdot 43^{2} + \left(33 a + 8\right)\cdot 43^{3} + \left(13 a + 10\right)\cdot 43^{4} + \left(28 a + 9\right)\cdot 43^{5} + \left(40 a + 21\right)\cdot 43^{6} + \left(39 a + 17\right)\cdot 43^{7} + \left(8 a + 20\right)\cdot 43^{8} + \left(41 a + 9\right)\cdot 43^{9} + \left(17 a + 9\right)\cdot 43^{10} + \left(2 a + 1\right)\cdot 43^{11} + \left(24 a + 31\right)\cdot 43^{12} +O\left(43^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 + 26\cdot 43 + 10\cdot 43^{2} + 10\cdot 43^{3} + 17\cdot 43^{4} + 26\cdot 43^{5} + 31\cdot 43^{6} + 8\cdot 43^{7} + 23\cdot 43^{8} + 14\cdot 43^{9} + 26\cdot 43^{10} + 12\cdot 43^{11} + 9\cdot 43^{12} +O\left(43^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a + 6 + \left(6 a + 26\right)\cdot 43 + 10\cdot 43^{2} + \left(9 a + 42\right)\cdot 43^{3} + \left(29 a + 32\right)\cdot 43^{4} + \left(14 a + 23\right)\cdot 43^{5} + \left(2 a + 33\right)\cdot 43^{6} + \left(3 a + 16\right)\cdot 43^{7} + \left(34 a + 32\right)\cdot 43^{8} + \left(a + 41\right)\cdot 43^{9} + \left(25 a + 28\right)\cdot 43^{10} + \left(40 a + 28\right)\cdot 43^{11} + \left(18 a + 9\right)\cdot 43^{12} +O\left(43^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 a + 27 + \left(6 a + 20\right)\cdot 43 + 38\cdot 43^{2} + \left(9 a + 34\right)\cdot 43^{3} + \left(29 a + 32\right)\cdot 43^{4} + \left(14 a + 33\right)\cdot 43^{5} + \left(2 a + 21\right)\cdot 43^{6} + \left(3 a + 25\right)\cdot 43^{7} + \left(34 a + 22\right)\cdot 43^{8} + \left(a + 33\right)\cdot 43^{9} + \left(25 a + 33\right)\cdot 43^{10} + \left(40 a + 41\right)\cdot 43^{11} + \left(18 a + 11\right)\cdot 43^{12} +O\left(43^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,5)$ |
| $(1,4)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-3$ |
| $3$ |
$2$ |
$(1,4)$ |
$1$ |
| $3$ |
$2$ |
$(1,4)(2,5)$ |
$-1$ |
| $6$ |
$2$ |
$(2,3)(5,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,4)(2,3)(5,6)$ |
$1$ |
| $8$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,5,4,2)$ |
$-1$ |
| $6$ |
$4$ |
$(1,4)(2,6,5,3)$ |
$1$ |
| $8$ |
$6$ |
$(1,5,6,4,2,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
