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:
| $r_{ 1 }$ |
$=$ |
$ 11 a^{2} + 31 a + 38 + \left(28 a^{2} + 39 a + 13\right)\cdot 43 + \left(26 a^{2} + 4 a + 31\right)\cdot 43^{2} + \left(35 a^{2} + 28 a + 22\right)\cdot 43^{3} + \left(15 a^{2} + 35 a + 31\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 41 a^{2} + 24 a + 24 + \left(21 a^{2} + 17\right)\cdot 43 + \left(26 a^{2} + 9 a + 36\right)\cdot 43^{2} + \left(16 a^{2} + 28 a + 41\right)\cdot 43^{3} + \left(42 a^{2} + 18 a + 27\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a^{2} + 32 a + 23 + \left(35 a^{2} + 37 a + 26\right)\cdot 43 + \left(39 a^{2} + 8 a + 16\right)\cdot 43^{2} + \left(9 a^{2} + 41 a + 37\right)\cdot 43^{3} + \left(27 a^{2} + 33 a + 17\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 + 15\cdot 43 + 40\cdot 43^{2} + 43^{3} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 36 + 33\cdot 43 + 34\cdot 43^{2} + 37\cdot 43^{3} + 23\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 34 a^{2} + 5 a + 39 + \left(19 a^{2} + 3 a + 36\right)\cdot 43 + \left(14 a^{2} + 24 a + 8\right)\cdot 43^{2} + \left(10 a + 42\right)\cdot 43^{3} + \left(30 a^{2} + 39 a + 40\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 41 a^{2} + 7 a + 15 + \left(37 a^{2} + 20\right)\cdot 43 + \left(a^{2} + 14 a\right)\cdot 43^{2} + \left(7 a^{2} + 4 a + 18\right)\cdot 43^{3} + \left(40 a^{2} + 11 a + 33\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 27 a^{2} + 30 a + 29 + \left(28 a^{2} + 4 a + 7\right)\cdot 43 + \left(19 a^{2} + 25 a + 3\right)\cdot 43^{2} + \left(16 a^{2} + 16 a + 13\right)\cdot 43^{3} + \left(16 a^{2} + 33 a + 39\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,6,7)(2,8,3)$ |
| $(1,7,8,2)(3,5,6,4)$ |
| $(1,4,8,5)(2,3,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $3$ | $(2,6,4)(3,5,7)$ | $\zeta_{3} + 1$ |
| $4$ | $3$ | $(2,4,6)(3,7,5)$ | $-\zeta_{3}$ |
| $6$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $4$ | $6$ | $(1,3,7,8,6,2)(4,5)$ | $-\zeta_{3} - 1$ |
| $4$ | $6$ | $(1,2,6,8,7,3)(4,5)$ | $\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
