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:
| $r_{ 1 }$ |
$=$ |
$ 23 a + 40 + \left(35 a + 26\right)\cdot 43 + \left(3 a + 2\right)\cdot 43^{2} + \left(9 a + 1\right)\cdot 43^{3} + \left(10 a + 25\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 a + 33 + \left(35 a + 4\right)\cdot 43 + \left(32 a + 18\right)\cdot 43^{2} + \left(21 a + 33\right)\cdot 43^{3} + \left(33 a + 5\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 a + 42 + \left(12 a + 32\right)\cdot 43 + \left(a + 11\right)\cdot 43^{2} + \left(2 a + 19\right)\cdot 43^{3} + \left(32 a + 20\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 a + 20 + \left(7 a + 39\right)\cdot 43 + \left(39 a + 13\right)\cdot 43^{2} + \left(33 a + 6\right)\cdot 43^{3} + \left(32 a + 26\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 36 + 36\cdot 43 + 23\cdot 43^{2} + 26\cdot 43^{3} + 26\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 18 a + 24 + \left(30 a + 20\right)\cdot 43 + 41 a\cdot 43^{2} + \left(40 a + 20\right)\cdot 43^{3} + \left(10 a + 7\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 13 a + 20 + \left(7 a + 10\right)\cdot 43 + \left(10 a + 15\right)\cdot 43^{2} + \left(21 a + 22\right)\cdot 43^{3} + \left(9 a + 17\right)\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.
