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