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:
| $r_{ 1 }$ |
$=$ |
$ 40 a + 22 + \left(34 a + 17\right)\cdot 43 + \left(27 a + 39\right)\cdot 43^{2} + \left(37 a + 35\right)\cdot 43^{3} + \left(16 a + 39\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 + 33\cdot 43 + 30\cdot 43^{2} + 4\cdot 43^{3} + 19\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 34 a + 3 + \left(23 a + 25\right)\cdot 43 + \left(28 a + 7\right)\cdot 43^{2} + \left(41 a + 20\right)\cdot 43^{3} + \left(34 a + 13\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 3 a + 19 + \left(8 a + 12\right)\cdot 43 + \left(15 a + 32\right)\cdot 43^{2} + \left(5 a + 2\right)\cdot 43^{3} + \left(26 a + 19\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a + 37 + \left(19 a + 14\right)\cdot 43 + \left(14 a + 12\right)\cdot 43^{2} + \left(a + 33\right)\cdot 43^{3} + \left(8 a + 6\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 24 + 25\cdot 43 + 6\cdot 43^{2} + 32\cdot 43^{3} + 30\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
