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:
| $r_{ 1 }$ |
$=$ |
$ 15 a + 15 + \left(16 a + 20\right)\cdot 41 + \left(37 a + 20\right)\cdot 41^{2} + \left(28 a + 25\right)\cdot 41^{3} + \left(33 a + 20\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 a + 16 + \left(33 a + 20\right)\cdot 41 + \left(4 a + 12\right)\cdot 41^{2} + \left(36 a + 30\right)\cdot 41^{3} + \left(15 a + 1\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a + 40 + \left(10 a + 22\right)\cdot 41 + 15\cdot 41^{2} + \left(24 a + 16\right)\cdot 41^{3} + \left(5 a + 21\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 a + 19 + \left(24 a + 13\right)\cdot 41 + \left(3 a + 34\right)\cdot 41^{2} + \left(12 a + 33\right)\cdot 41^{3} + \left(7 a + 10\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 a + 24 + \left(7 a + 9\right)\cdot 41 + \left(36 a + 34\right)\cdot 41^{2} + \left(4 a + 10\right)\cdot 41^{3} + \left(25 a + 13\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 a + 12 + \left(30 a + 36\right)\cdot 41 + \left(40 a + 5\right)\cdot 41^{2} + \left(16 a + 6\right)\cdot 41^{3} + \left(35 a + 14\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(2,6)(3,4)$ |
| $(4,5,6)$ |
| $(1,3,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $3$ |
$2$ |
$(1,5)(2,6)(3,4)$ |
$0$ |
$0$ |
| $1$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$2 \zeta_{3}$ |
$-2 \zeta_{3} - 2$ |
| $1$ |
$3$ |
$(1,3,2)(4,6,5)$ |
$-2 \zeta_{3} - 2$ |
$2 \zeta_{3}$ |
| $2$ |
$3$ |
$(1,3,2)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $2$ |
$3$ |
$(1,2,3)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $2$ |
$3$ |
$(1,3,2)(4,5,6)$ |
$-1$ |
$-1$ |
| $3$ |
$6$ |
$(1,6,2,4,3,5)$ |
$0$ |
$0$ |
| $3$ |
$6$ |
$(1,5,3,4,2,6)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
