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:
| $r_{ 1 }$ |
$=$ |
$ 10 + 26\cdot 41 + 20\cdot 41^{2} + 16\cdot 41^{3} + 36\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 a + 29 + \left(36 a + 31\right)\cdot 41 + \left(39 a + 12\right)\cdot 41^{2} + 40 a\cdot 41^{3} + \left(11 a + 13\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 a + 35 + \left(4 a + 14\right)\cdot 41 + \left(a + 14\right)\cdot 41^{2} + 41^{3} + \left(29 a + 8\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 a + 20 + \left(21 a + 31\right)\cdot 41 + \left(37 a + 4\right)\cdot 41^{2} + 14 a\cdot 41^{3} + \left(4 a + 33\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 26 + 39\cdot 41 + 15\cdot 41^{2} + 15\cdot 41^{3} + 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 5 a + 5 + \left(19 a + 20\right)\cdot 41 + \left(3 a + 13\right)\cdot 41^{2} + \left(26 a + 7\right)\cdot 41^{3} + \left(36 a + 31\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3)(4,6)$ |
| $(1,2)(5,6)$ |
| $(1,4)(2,6)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,4)$ |
$-2$ |
| $3$ |
$2$ |
$(1,2)(5,6)$ |
$0$ |
| $3$ |
$2$ |
$(1,4)(2,6)(3,5)$ |
$0$ |
| $2$ |
$3$ |
$(1,3,2)(4,6,5)$ |
$-1$ |
| $2$ |
$6$ |
$(1,6,3,5,2,4)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
