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:
| $r_{ 1 }$ |
$=$ |
$ 3 a + 14 + \left(20 a + 3\right)\cdot 41 + \left(13 a + 39\right)\cdot 41^{2} + \left(38 a + 40\right)\cdot 41^{3} + \left(16 a + 22\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 + 27\cdot 41 + 29\cdot 41^{2} + 3\cdot 41^{3} + 32\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 a + \left(2 a + 30\right)\cdot 41 + \left(38 a + 35\right)\cdot 41^{2} + \left(40 a + 35\right)\cdot 41^{3} + \left(35 a + 25\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a + 23 + \left(38 a + 3\right)\cdot 41 + \left(2 a + 24\right)\cdot 41^{2} + 38\cdot 41^{3} + \left(5 a + 10\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 + 39\cdot 41 + 16\cdot 41^{2} + 25\cdot 41^{3} + 36\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 38 a + 23 + \left(20 a + 19\right)\cdot 41 + \left(27 a + 18\right)\cdot 41^{2} + \left(2 a + 19\right)\cdot 41^{3} + \left(24 a + 35\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$10$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$2$ |
| $15$ |
$2$ |
$(1,2)$ |
$-2$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$-2$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$-1$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
