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:
| $r_{ 1 }$ |
$=$ |
$ 15 a + 17 + \left(18 a + 34\right)\cdot 41 + \left(24 a + 20\right)\cdot 41^{2} + \left(33 a + 8\right)\cdot 41^{3} + \left(36 a + 28\right)\cdot 41^{4} + \left(38 a + 34\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 + 31\cdot 41 + 4\cdot 41^{2} + 27\cdot 41^{3} + 18\cdot 41^{4} + 27\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 38 + \left(9 a + 4\right)\cdot 41 + \left(14 a + 2\right)\cdot 41^{2} + \left(18 a + 34\right)\cdot 41^{3} + \left(32 a + 36\right)\cdot 41^{4} + \left(28 a + 14\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 37 + 33\cdot 41 + 24\cdot 41^{2} + 16\cdot 41^{3} + 23\cdot 41^{4} + 26\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 26 a + 21 + \left(22 a + 33\right)\cdot 41 + \left(16 a + 34\right)\cdot 41^{2} + \left(7 a + 2\right)\cdot 41^{3} + \left(4 a + 23\right)\cdot 41^{4} + \left(2 a + 32\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 34 a + 18 + \left(31 a + 25\right)\cdot 41 + \left(26 a + 35\right)\cdot 41^{2} + \left(22 a + 33\right)\cdot 41^{3} + \left(8 a + 33\right)\cdot 41^{4} + \left(12 a + 27\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4)$ |
| $(1,5)$ |
| $(3,6)$ |
| $(1,3,2)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,5)(2,4)(3,6)$ |
$-3$ |
| $3$ |
$2$ |
$(3,6)$ |
$1$ |
| $3$ |
$2$ |
$(1,5)(3,6)$ |
$-1$ |
| $4$ |
$3$ |
$(1,3,2)(4,5,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,3)(4,6,5)$ |
$0$ |
| $4$ |
$6$ |
$(1,3,4,5,6,2)$ |
$0$ |
| $4$ |
$6$ |
$(1,2,6,5,4,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
