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:
| $r_{ 1 }$ |
$=$ |
$ 33 a + 1 + \left(9 a + 39\right)\cdot 41 + \left(11 a + 26\right)\cdot 41^{2} + \left(16 a + 22\right)\cdot 41^{3} + \left(38 a + 12\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 a + 10 + \left(16 a + 21\right)\cdot 41 + \left(39 a + 27\right)\cdot 41^{2} + \left(16 a + 31\right)\cdot 41^{3} + \left(32 a + 38\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 8 a + 18 + \left(31 a + 35\right)\cdot 41 + \left(29 a + 9\right)\cdot 41^{2} + \left(24 a + 19\right)\cdot 41^{3} + \left(2 a + 29\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 a + 16 + \left(24 a + 26\right)\cdot 41 + \left(a + 6\right)\cdot 41^{2} + \left(24 a + 2\right)\cdot 41^{3} + \left(8 a + 37\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 + 7\cdot 41 + 4\cdot 41^{2} + 40\cdot 41^{3} + 39\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 15 + 34\cdot 41 + 6\cdot 41^{2} + 7\cdot 41^{3} + 6\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,6)(3,5)$ |
| $(1,2,5,4,3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-2$ |
| $3$ | $2$ | $(2,6)(3,5)$ | $0$ |
| $3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,5,3)(2,4,6)$ | $-1$ |
| $2$ | $6$ | $(1,2,5,4,3,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
