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:
| $r_{ 1 }$ |
$=$ |
$ 34 a + 7 + \left(36 a + 12\right)\cdot 43 + \left(42 a + 4\right)\cdot 43^{2} + \left(26 a + 6\right)\cdot 43^{3} + \left(10 a + 23\right)\cdot 43^{4} + \left(13 a + 12\right)\cdot 43^{5} + 6\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 a + 17 + \left(a + 40\right)\cdot 43 + \left(19 a + 39\right)\cdot 43^{2} + \left(25 a + 40\right)\cdot 43^{3} + \left(29 a + 30\right)\cdot 43^{4} + \left(23 a + 22\right)\cdot 43^{5} + \left(4 a + 19\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 a + 37 + \left(41 a + 21\right)\cdot 43 + \left(23 a + 14\right)\cdot 43^{2} + \left(17 a + 4\right)\cdot 43^{3} + \left(13 a + 35\right)\cdot 43^{4} + \left(19 a + 16\right)\cdot 43^{5} + 38 a\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 a + 41 + \left(6 a + 14\right)\cdot 43 + 10\cdot 43^{2} + \left(16 a + 33\right)\cdot 43^{3} + \left(32 a + 6\right)\cdot 43^{4} + \left(29 a + 15\right)\cdot 43^{5} + \left(42 a + 36\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 a + \left(32 a + 11\right)\cdot 43 + \left(42 a + 31\right)\cdot 43^{2} + \left(25 a + 36\right)\cdot 43^{3} + \left(16 a + 39\right)\cdot 43^{4} + \left(40 a + 2\right)\cdot 43^{5} + \left(5 a + 10\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 26 a + 17 + \left(10 a + 26\right)\cdot 43 + 41\cdot 43^{2} + \left(17 a + 19\right)\cdot 43^{3} + \left(26 a + 30\right)\cdot 43^{4} + \left(2 a + 26\right)\cdot 43^{5} + \left(37 a + 18\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 24 + 26\cdot 43 + 20\cdot 43^{2} + 23\cdot 43^{3} + 40\cdot 43^{4} + 32\cdot 43^{5} + 21\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 29 + 18\cdot 43 + 9\cdot 43^{2} + 7\cdot 43^{3} + 8\cdot 43^{4} + 42\cdot 43^{5} + 15\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7)(3,5)(4,8)$ |
| $(1,5,8)(3,7,4)$ |
| $(1,4)(2,6)(3,5)(7,8)$ |
| $(1,2,4,6)(3,7,5,8)$ |
| $(1,8,4,7)(2,5,6,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,6)(3,5)(7,8)$ | $-2$ |
| $12$ | $2$ | $(1,7)(3,5)(4,8)$ | $0$ |
| $8$ | $3$ | $(1,5,8)(3,7,4)$ | $-1$ |
| $6$ | $4$ | $(1,2,4,6)(3,7,5,8)$ | $0$ |
| $8$ | $6$ | $(1,8,2,4,7,6)(3,5)$ | $1$ |
| $6$ | $8$ | $(1,2,8,5,4,6,7,3)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,6,8,3,4,2,7,5)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
