Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 39 a + 5 + \left(4 a + 10\right)\cdot 41 + \left(3 a + 1\right)\cdot 41^{2} + \left(22 a + 2\right)\cdot 41^{3} + \left(16 a + 6\right)\cdot 41^{4} + \left(37 a + 16\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 a + 40 + \left(36 a + 26\right)\cdot 41 + \left(37 a + 5\right)\cdot 41^{2} + \left(18 a + 24\right)\cdot 41^{3} + \left(24 a + 33\right)\cdot 41^{4} + \left(3 a + 29\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 + 13\cdot 41 + 29\cdot 41^{2} + 13\cdot 41^{3} + 40\cdot 41^{4} + 16\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a + 37 + \left(36 a + 30\right)\cdot 41 + \left(37 a + 39\right)\cdot 41^{2} + \left(18 a + 38\right)\cdot 41^{3} + \left(24 a + 34\right)\cdot 41^{4} + \left(3 a + 24\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 39 a + 2 + \left(4 a + 14\right)\cdot 41 + \left(3 a + 35\right)\cdot 41^{2} + \left(22 a + 16\right)\cdot 41^{3} + \left(16 a + 7\right)\cdot 41^{4} + \left(37 a + 11\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 + 27\cdot 41 + 11\cdot 41^{2} + 27\cdot 41^{3} + 24\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,5)$ |
| $(1,3)(2,5)(4,6)$ |
| $(2,6)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,6,2)(3,5,4)$ | $-1$ |
| $2$ | $6$ | $(1,5,6,4,2,3)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
