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:
| $r_{ 1 }$ |
$=$ |
$ 30 a + 40 + \left(13 a^{2} + 3 a + 4\right)\cdot 43 + \left(30 a^{2} + 22 a + 21\right)\cdot 43^{2} + \left(21 a^{2} + 21 a\right)\cdot 43^{3} + \left(30 a^{2} + 18\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 38 a^{2} + 5 a + 15 + \left(16 a^{2} + 34 a + 3\right)\cdot 43 + \left(26 a^{2} + 23 a + 2\right)\cdot 43^{2} + \left(14 a^{2} + 11 a + 32\right)\cdot 43^{3} + \left(13 a^{2} + 15 a + 5\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 40 + 34\cdot 43 + 18\cdot 43^{3} + 16\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 a^{2} + 21 a + 31 + \left(24 a^{2} + 20 a + 26\right)\cdot 43 + \left(31 a + 15\right)\cdot 43^{2} + \left(33 a^{2} + 13 a + 22\right)\cdot 43^{3} + \left(24 a^{2} + 4 a + 28\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 a^{2} + 15 a + 1 + \left(12 a^{2} + 14 a + 29\right)\cdot 43 + \left(26 a^{2} + 3 a + 30\right)\cdot 43^{2} + \left(27 a^{2} + 14 a + 40\right)\cdot 43^{3} + \left(31 a^{2} + 40 a + 17\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 a^{2} + 35 a + 6 + \left(5 a^{2} + 18 a\right)\cdot 43 + \left(12 a^{2} + 32 a + 9\right)\cdot 43^{2} + \left(31 a^{2} + 7 a + 21\right)\cdot 43^{3} + \left(30 a^{2} + 38 a + 32\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 31 a^{2} + 23 a + 39 + \left(13 a^{2} + 37 a + 29\right)\cdot 43 + \left(33 a^{2} + 15 a + 6\right)\cdot 43^{2} + \left(17 a + 37\right)\cdot 43^{3} + \left(41 a^{2} + 30 a + 9\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,7)(2,4)$ |
| $(2,3,5,7)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$7$ |
| $21$ |
$2$ |
$(1,7)(2,4)$ |
$-1$ |
| $56$ |
$3$ |
$(1,2,5)(3,7,6)$ |
$1$ |
| $42$ |
$4$ |
$(2,3,5,7)(4,6)$ |
$-1$ |
| $24$ |
$7$ |
$(1,7,4,6,2,3,5)$ |
$0$ |
| $24$ |
$7$ |
$(1,6,5,4,3,7,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
