Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 a + 37 + \left(17 a + 42\right)\cdot 43 + \left(29 a + 11\right)\cdot 43^{2} + 16 a\cdot 43^{3} + \left(2 a + 1\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 a + 2 + \left(10 a + 26\right)\cdot 43 + \left(34 a + 12\right)\cdot 43^{2} + \left(12 a + 4\right)\cdot 43^{3} + \left(2 a + 18\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 37\cdot 43 + 21\cdot 43^{2} + 24\cdot 43^{3} + 4\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 a + 7 + \left(32 a + 31\right)\cdot 43 + \left(8 a + 36\right)\cdot 43^{2} + \left(30 a + 25\right)\cdot 43^{3} + \left(40 a + 7\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 39 a + 41 + \left(25 a + 12\right)\cdot 43 + \left(13 a + 24\right)\cdot 43^{2} + \left(26 a + 30\right)\cdot 43^{3} + \left(40 a + 29\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 + 21\cdot 43 + 21\cdot 43^{2} + 25\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(3,5)$ |
| $(1,5,4)(2,6,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $15$ |
$2$ |
$(1,4)(3,5)$ |
$-1$ |
$-1$ |
| $20$ |
$3$ |
$(1,5,4)(2,6,3)$ |
$0$ |
$0$ |
| $12$ |
$5$ |
$(2,6,3,4,5)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ |
$5$ |
$(2,3,5,6,4)$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
