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:
| $r_{ 1 }$ |
$=$ |
$ 6 + 18\cdot 41 + 12\cdot 41^{2} + 15\cdot 41^{3} + 26\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 36 + 22\cdot 41 + 28\cdot 41^{2} + 25\cdot 41^{3} + 14\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ a + 37 + \left(36 a + 33\right)\cdot 41 + \left(a + 27\right)\cdot 41^{2} + \left(38 a + 20\right)\cdot 41^{3} + \left(2 a + 29\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a + 2 + \left(36 a + 23\right)\cdot 41 + \left(a + 2\right)\cdot 41^{2} + \left(38 a + 31\right)\cdot 41^{3} + \left(2 a + 40\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 a + 5 + \left(4 a + 7\right)\cdot 41 + \left(39 a + 13\right)\cdot 41^{2} + \left(2 a + 20\right)\cdot 41^{3} + \left(38 a + 11\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 40 a + 40 + \left(4 a + 17\right)\cdot 41 + \left(39 a + 38\right)\cdot 41^{2} + \left(2 a + 9\right)\cdot 41^{3} + 38 a\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(3,6)(4,5)$ |
| $(1,3,4,2,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-2$ |
| $3$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,4,5)(2,6,3)$ | $-1$ |
| $2$ | $6$ | $(1,3,4,2,5,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
