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^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ a + 15 + \left(2 a + 2\right)\cdot 43 + \left(13 a + 8\right)\cdot 43^{2} + \left(11 a + 10\right)\cdot 43^{3} + \left(9 a + 21\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 16 + \left(2 a + 37\right)\cdot 43 + \left(9 a + 13\right)\cdot 43^{2} + 43^{3} + \left(4 a + 34\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 42 a + 16 + \left(40 a + 3\right)\cdot 43 + \left(29 a + 19\right)\cdot 43^{2} + \left(31 a + 8\right)\cdot 43^{3} + \left(33 a + 19\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 + 15\cdot 43 + 35\cdot 43^{2} + 33\cdot 43^{3} + 38\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 2 + 19\cdot 43 + 21\cdot 43^{2} + 23\cdot 43^{3} + 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 31 a + 28 + \left(40 a + 27\right)\cdot 43 + \left(33 a + 20\right)\cdot 43^{2} + \left(42 a + 35\right)\cdot 43^{3} + \left(38 a + 37\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 29 + 23\cdot 43 + 10\cdot 43^{2} + 16\cdot 43^{3} + 19\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $15$ |
| $21$ | $2$ | $(1,2)$ | $5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
