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:
| $r_{ 1 }$ |
$=$ |
$ 3 + 7\cdot 41 + 15\cdot 41^{2} + 36\cdot 41^{3} + 29\cdot 41^{4} + 20\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 a + 24 + \left(4 a + 5\right)\cdot 41 + \left(26 a + 39\right)\cdot 41^{2} + \left(24 a + 4\right)\cdot 41^{3} + \left(12 a + 8\right)\cdot 41^{4} + \left(3 a + 14\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 a + 14 + \left(11 a + 3\right)\cdot 41 + \left(11 a + 14\right)\cdot 41^{2} + \left(34 a + 14\right)\cdot 41^{3} + \left(40 a + 4\right)\cdot 41^{4} + \left(8 a + 5\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 a + 30 + \left(36 a + 15\right)\cdot 41 + \left(14 a + 31\right)\cdot 41^{2} + \left(16 a + 11\right)\cdot 41^{3} + \left(28 a + 21\right)\cdot 41^{4} + \left(37 a + 11\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 + 28\cdot 41 + 27\cdot 41^{2} + 31\cdot 41^{3} + 7\cdot 41^{4} + 39\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 a + 21 + \left(29 a + 21\right)\cdot 41 + \left(29 a + 36\right)\cdot 41^{2} + \left(6 a + 23\right)\cdot 41^{3} + 10\cdot 41^{4} + \left(32 a + 32\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(5,6)$ |
| $(3,6,5)$ |
| $(1,2,4)(3,6,5)$ |
| $(2,4)(5,6)$ |
| $(1,5,4,3,2,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $3$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
| $3$ | $2$ | $(1,6)(2,5)(3,4)$ | $0$ |
| $9$ | $2$ | $(1,4)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,2,4)(3,6,5)$ | $-2$ |
| $2$ | $3$ | $(1,4,2)(3,6,5)$ | $-2$ |
| $4$ | $3$ | $(1,4,2)$ | $1$ |
| $6$ | $6$ | $(1,5,4,3,2,6)$ | $0$ |
| $6$ | $6$ | $(1,3,2,6,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
