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^{3} + x + 40 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 5\cdot 43 + 41\cdot 43^{2} + 34\cdot 43^{3} + 38\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 41 a^{2} + 35 a + 13 + \left(22 a^{2} + 17 a\right)\cdot 43 + \left(34 a^{2} + 6 a + 15\right)\cdot 43^{2} + \left(8 a^{2} + 3 a + 21\right)\cdot 43^{3} + \left(15 a^{2} + 22 a + 6\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 a^{2} + 8 a + 8 + \left(23 a^{2} + 2 a\right)\cdot 43 + \left(26 a^{2} + 9 a + 12\right)\cdot 43^{2} + \left(2 a^{2} + 9 a + 3\right)\cdot 43^{3} + \left(31 a^{2} + 27 a + 40\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 a^{2} + 9 a + 11 + \left(7 a^{2} + a + 33\right)\cdot 43 + \left(5 a^{2} + 4 a + 9\right)\cdot 43^{2} + \left(a^{2} + 22 a + 16\right)\cdot 43^{3} + \left(17 a^{2} + 31 a + 36\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 a^{2} + 42 a + 19 + \left(12 a^{2} + 23 a + 7\right)\cdot 43 + \left(3 a^{2} + 32 a + 37\right)\cdot 43^{2} + \left(33 a^{2} + 17 a + 8\right)\cdot 43^{3} + \left(10 a^{2} + 32 a + 32\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 32 a^{2} + 6 a + 33 + \left(23 a^{2} + 22 a + 14\right)\cdot 43 + \left(30 a^{2} + 33 a\right)\cdot 43^{2} + \left(38 a^{2} + 33 a + 13\right)\cdot 43^{3} + \left(14 a^{2} + 32 a + 29\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 38 a^{2} + 29 a + 37 + \left(38 a^{2} + 18 a + 24\right)\cdot 43 + \left(28 a^{2} + 13\right)\cdot 43^{2} + \left(a^{2} + 31\right)\cdot 43^{3} + \left(40 a^{2} + 26 a + 31\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,3,6,4)(5,7)$ |
| $(1,2)(3,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)(3,7)$ | $2$ |
| $56$ | $3$ | $(1,5,6)(2,3,4)$ | $0$ |
| $42$ | $4$ | $(1,3,6,4)(5,7)$ | $0$ |
| $24$ | $7$ | $(1,7,5,3,6,4,2)$ | $-1$ |
| $24$ | $7$ | $(1,3,2,5,4,7,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
