Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{3} + x + 40 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 23 a^{2} + 38 a + 1 + \left(34 a^{2} + 35 a + 23\right)\cdot 43 + \left(11 a^{2} + 5 a + 36\right)\cdot 43^{2} + \left(24 a + 28\right)\cdot 43^{3} + \left(14 a^{2} + 15 a + 23\right)\cdot 43^{4} + \left(33 a^{2} + 22 a + 36\right)\cdot 43^{5} + \left(8 a^{2} + 16 a + 5\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 a^{2} + 21 a + 17 + \left(36 a^{2} + 24 a + 38\right)\cdot 43 + \left(18 a^{2} + 11 a + 26\right)\cdot 43^{2} + \left(11 a^{2} + 29 a + 7\right)\cdot 43^{3} + \left(23 a^{2} + 23 a + 1\right)\cdot 43^{4} + \left(15 a^{2} + 3 a + 39\right)\cdot 43^{5} + \left(32 a^{2} + 35\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 a^{2} + 27 a + 25 + \left(15 a^{2} + 25 a + 24\right)\cdot 43 + \left(12 a^{2} + 25 a + 22\right)\cdot 43^{2} + \left(31 a^{2} + 32 a + 6\right)\cdot 43^{3} + \left(5 a^{2} + 3 a + 18\right)\cdot 43^{4} + \left(37 a^{2} + 17 a + 10\right)\cdot 43^{5} + \left(a^{2} + 26 a + 1\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 a^{2} + 29 a + 11 + \left(39 a^{2} + 2 a + 12\right)\cdot 43 + \left(42 a^{2} + 23 a + 14\right)\cdot 43^{2} + \left(32 a^{2} + 26 a + 36\right)\cdot 43^{3} + \left(38 a^{2} + 40 a + 25\right)\cdot 43^{4} + \left(34 a^{2} + 34 a + 37\right)\cdot 43^{5} + \left(34 a^{2} + a + 8\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 a^{2} + 8 a + 25 + \left(38 a^{2} + 10 a + 25\right)\cdot 43 + \left(42 a^{2} + 10 a + 28\right)\cdot 43^{2} + \left(29 a^{2} + 40 a + 5\right)\cdot 43^{3} + \left(37 a^{2} + 42 a + 25\right)\cdot 43^{4} + \left(27 a^{2} + 24 a + 18\right)\cdot 43^{5} + \left(35 a^{2} + 24 a + 9\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 + 40\cdot 43 + 5\cdot 43^{2} + 11\cdot 43^{3} + 20\cdot 43^{4} + 24\cdot 43^{5} + 34\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 13 + 17\cdot 43 + 28\cdot 43^{2} + 39\cdot 43^{3} + 33\cdot 43^{4} + 25\cdot 43^{5} + 10\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 32 a^{2} + 6 a + 7 + \left(7 a^{2} + 30 a + 5\right)\cdot 43 + 9 a\cdot 43^{2} + \left(23 a^{2} + 19 a + 1\right)\cdot 43^{3} + \left(9 a^{2} + 2 a + 35\right)\cdot 43^{4} + \left(23 a^{2} + 26 a + 29\right)\cdot 43^{5} + \left(15 a^{2} + 16 a + 24\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 24 + 28\cdot 43 + 8\cdot 43^{2} + 35\cdot 43^{3} + 31\cdot 43^{4} + 35\cdot 43^{5} + 40\cdot 43^{6} +O\left(43^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,4,6)(2,5,9)(3,8,7)$ |
| $(4,5,8)(6,7,9)$ |
| $(1,2,3)(4,8,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $3$ | $(1,3,2)(4,8,5)(6,7,9)$ | $3 \zeta_{3}$ |
| $1$ | $3$ | $(1,2,3)(4,5,8)(6,9,7)$ | $-3 \zeta_{3} - 3$ |
| $3$ | $3$ | $(1,4,6)(2,5,9)(3,8,7)$ | $0$ |
| $3$ | $3$ | $(1,6,4)(2,9,5)(3,7,8)$ | $0$ |
| $3$ | $3$ | $(1,2,3)(4,8,5)$ | $0$ |
| $3$ | $3$ | $(1,3,2)(4,5,8)$ | $0$ |
| $3$ | $3$ | $(1,8,7)(2,4,6)(3,5,9)$ | $0$ |
| $3$ | $3$ | $(1,7,8)(2,6,4)(3,9,5)$ | $0$ |
| $3$ | $3$ | $(1,9,5)(2,7,8)(3,6,4)$ | $0$ |
| $3$ | $3$ | $(1,5,9)(2,8,7)(3,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
