Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 20 a + 4 + \left(11 a + 21\right)\cdot 41 + \left(37 a + 14\right)\cdot 41^{2} + 33\cdot 41^{3} + \left(2 a + 1\right)\cdot 41^{4} + \left(27 a + 7\right)\cdot 41^{5} + \left(38 a + 24\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 a + 33 + 29 a\cdot 41 + \left(3 a + 29\right)\cdot 41^{2} + \left(40 a + 23\right)\cdot 41^{3} + \left(38 a + 5\right)\cdot 41^{4} + \left(13 a + 21\right)\cdot 41^{5} + \left(2 a + 5\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 + 41 + 16\cdot 41^{2} + 37\cdot 41^{3} + 32\cdot 41^{4} + 19\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 + 36\cdot 41 + 19\cdot 41^{2} + 12\cdot 41^{3} + 34\cdot 41^{4} + 24\cdot 41^{5} + 3\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 a + 14 + \left(11 a + 27\right)\cdot 41 + \left(37 a + 10\right)\cdot 41^{2} + 17\cdot 41^{3} + 2 a\cdot 41^{4} + \left(27 a + 24\right)\cdot 41^{5} + \left(38 a + 39\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 a + 23 + \left(29 a + 35\right)\cdot 41 + \left(3 a + 32\right)\cdot 41^{2} + \left(40 a + 39\right)\cdot 41^{3} + \left(38 a + 6\right)\cdot 41^{4} + \left(13 a + 4\right)\cdot 41^{5} + \left(2 a + 31\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(2,3)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,4)$ |
$-2$ |
| $3$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $3$ |
$2$ |
$(1,4)(3,5)$ |
$0$ |
| $2$ |
$3$ |
$(1,6,4)(2,3,5)$ |
$-1$ |
| $2$ |
$6$ |
$(1,3,6,5,4,2)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
