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:
| $r_{ 1 }$ |
$=$ |
$ 11 + 5\cdot 41 + 31\cdot 41^{2} + 22\cdot 41^{3} + 19\cdot 41^{4} + 27\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 a + 13 + \left(35 a + 25\right)\cdot 41 + \left(38 a + 13\right)\cdot 41^{2} + \left(29 a + 30\right)\cdot 41^{3} + \left(34 a + 39\right)\cdot 41^{4} + \left(2 a + 40\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a + 40 + \left(20 a + 6\right)\cdot 41 + \left(22 a + 23\right)\cdot 41^{2} + \left(21 a + 18\right)\cdot 41^{3} + \left(25 a + 33\right)\cdot 41^{4} + \left(17 a + 20\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 + 5\cdot 41 + 13\cdot 41^{2} + 32\cdot 41^{3} + 32\cdot 41^{4} + 11\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a + 32 + \left(5 a + 29\right)\cdot 41 + \left(2 a + 12\right)\cdot 41^{2} + \left(11 a + 40\right)\cdot 41^{3} + \left(6 a + 31\right)\cdot 41^{4} + \left(38 a + 14\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 a + 12 + \left(20 a + 9\right)\cdot 41 + \left(18 a + 29\right)\cdot 41^{2} + \left(19 a + 19\right)\cdot 41^{3} + \left(15 a + 6\right)\cdot 41^{4} + \left(23 a + 7\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)$ |
| $(1,4)$ |
| $(3,6)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $3$ | $2$ | $(1,4)(2,5)$ | $-1$ |
| $4$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $4$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $4$ | $6$ | $(1,5,6,4,2,3)$ | $0$ |
| $4$ | $6$ | $(1,3,2,4,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
