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:
| $r_{ 1 }$ |
$=$ |
$ 38 a + 20 + \left(42 a + 5\right)\cdot 43 + \left(16 a + 23\right)\cdot 43^{2} + \left(31 a + 8\right)\cdot 43^{3} + \left(42 a + 7\right)\cdot 43^{4} + \left(36 a + 3\right)\cdot 43^{5} + 16\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 41 + 21\cdot 43 + 3\cdot 43^{2} + 25\cdot 43^{3} + 42\cdot 43^{4} + 31\cdot 43^{5} + 17\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 + 8\cdot 43 + 5\cdot 43^{2} + 15\cdot 43^{3} + 7\cdot 43^{4} + 8\cdot 43^{5} + 4\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 a + 38 + \left(20 a + 36\right)\cdot 43 + \left(33 a + 39\right)\cdot 43^{2} + \left(18 a + 25\right)\cdot 43^{3} + \left(20 a + 35\right)\cdot 43^{4} + \left(9 a + 37\right)\cdot 43^{5} + \left(38 a + 25\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 a + 27 + \left(a + 36\right)\cdot 43 + \left(8 a + 21\right)\cdot 43^{2} + \left(4 a + 33\right)\cdot 43^{3} + \left(20 a + 3\right)\cdot 43^{4} + \left(36 a + 25\right)\cdot 43^{5} + \left(6 a + 8\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 5 a + 15 + 10\cdot 43 + \left(26 a + 40\right)\cdot 43^{2} + \left(11 a + 22\right)\cdot 43^{3} + 18\cdot 43^{4} + \left(6 a + 40\right)\cdot 43^{5} + \left(42 a + 22\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 7 a + 31 + \left(22 a + 21\right)\cdot 43 + \left(9 a + 9\right)\cdot 43^{2} + \left(24 a + 11\right)\cdot 43^{3} + \left(22 a + 37\right)\cdot 43^{4} + \left(33 a + 26\right)\cdot 43^{5} + \left(4 a + 11\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 36 a + 34 + \left(41 a + 30\right)\cdot 43 + \left(34 a + 28\right)\cdot 43^{2} + \left(38 a + 29\right)\cdot 43^{3} + \left(22 a + 19\right)\cdot 43^{4} + \left(6 a + 41\right)\cdot 43^{5} + \left(36 a + 21\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6)(2,3)(4,8)(5,7)$ |
| $(2,7,4)(3,5,8)$ |
| $(1,4,6,8)(2,5,3,7)$ |
| $(1,5,6,7)(2,8,3,4)$ |
| $(2,8)(3,4)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,3)(4,8)(5,7)$ |
$-2$ |
$-2$ |
| $12$ |
$2$ |
$(2,8)(3,4)(5,7)$ |
$0$ |
$0$ |
| $8$ |
$3$ |
$(1,2,8)(3,4,6)$ |
$-1$ |
$-1$ |
| $6$ |
$4$ |
$(1,4,6,8)(2,5,3,7)$ |
$0$ |
$0$ |
| $8$ |
$6$ |
$(1,5,3,6,7,2)(4,8)$ |
$1$ |
$1$ |
| $6$ |
$8$ |
$(1,3,5,4,6,2,7,8)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ |
$8$ |
$(1,2,5,8,6,3,7,4)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
