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:
| $r_{ 1 }$ |
$=$ |
$ 13 a + 13 + \left(26 a + 20\right)\cdot 43 + \left(22 a + 30\right)\cdot 43^{2} + \left(9 a + 31\right)\cdot 43^{3} + \left(20 a + 15\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 a + 9 + \left(26 a + 41\right)\cdot 43 + \left(22 a + 17\right)\cdot 43^{2} + \left(9 a + 1\right)\cdot 43^{3} + 20 a\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 a + 15 + \left(16 a + 39\right)\cdot 43 + \left(20 a + 17\right)\cdot 43^{2} + \left(33 a + 16\right)\cdot 43^{3} + 22 a\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 a + 2 + \left(26 a + 26\right)\cdot 43 + \left(22 a + 21\right)\cdot 43^{2} + \left(9 a + 29\right)\cdot 43^{3} + \left(20 a + 32\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 a + 22 + \left(16 a + 11\right)\cdot 43 + \left(20 a + 14\right)\cdot 43^{2} + \left(33 a + 31\right)\cdot 43^{3} + \left(22 a + 10\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 30 a + 26 + \left(16 a + 33\right)\cdot 43 + \left(20 a + 26\right)\cdot 43^{2} + \left(33 a + 18\right)\cdot 43^{3} + \left(22 a + 26\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,4,6,2,3)$ |
| $(1,6)(2,5)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,4)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,4,2)(3,5,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,2,4)(3,6,5)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,5,4,6,2,3)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,3,2,6,4,5)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
