Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 11 + 6\cdot 41 + 8\cdot 41^{2} + 11\cdot 41^{3} + 22\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 36 a + 36 + \left(13 a + 19\right)\cdot 41 + \left(3 a + 26\right)\cdot 41^{2} + \left(18 a + 16\right)\cdot 41^{3} + \left(16 a + 36\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 a + 3 + \left(16 a + 20\right)\cdot 41 + \left(3 a + 8\right)\cdot 41^{2} + \left(18 a + 5\right)\cdot 41^{3} + \left(9 a + 4\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 40 + 11\cdot 41 + 14\cdot 41^{2} + 7\cdot 41^{3} + 19\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a + 14 + \left(24 a + 39\right)\cdot 41 + \left(37 a + 1\right)\cdot 41^{2} + \left(22 a + 15\right)\cdot 41^{3} + \left(31 a + 14\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 5 a + 21 + \left(27 a + 25\right)\cdot 41 + \left(37 a + 22\right)\cdot 41^{2} + \left(22 a + 26\right)\cdot 41^{3} + \left(24 a + 26\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,3)(5,6)$ |
| $(1,3,4,2,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,4)(2,3)(5,6)$ | $-2$ |
| $15$ | $2$ | $(2,4)(5,6)$ | $0$ |
| $20$ | $3$ | $(1,4,5)(2,6,3)$ | $1$ |
| $30$ | $4$ | $(2,5,4,6)$ | $0$ |
| $24$ | $5$ | $(1,3,4,6,2)$ | $-1$ |
| $20$ | $6$ | $(1,3,4,2,5,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
