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:
| $r_{ 1 }$ |
$=$ |
$ 6 + 28\cdot 41 + 15\cdot 41^{3} + 38\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 37\cdot 41 + 28\cdot 41^{2} + 15\cdot 41^{3} + 23\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 a + 6 + \left(11 a + 11\right)\cdot 41 + \left(4 a + 20\right)\cdot 41^{2} + \left(39 a + 16\right)\cdot 41^{3} + \left(27 a + 25\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 a + 25 + 28 a\cdot 41 + \left(39 a + 1\right)\cdot 41^{2} + \left(11 a + 16\right)\cdot 41^{3} + \left(38 a + 13\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a + 25 + \left(29 a + 25\right)\cdot 41 + \left(36 a + 21\right)\cdot 41^{2} + \left(a + 6\right)\cdot 41^{3} + \left(13 a + 29\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 16 a + 18 + \left(12 a + 20\right)\cdot 41 + \left(a + 9\right)\cdot 41^{2} + \left(29 a + 12\right)\cdot 41^{3} + \left(2 a + 34\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,6)$ |
| $(1,4,2)(3,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $15$ |
$2$ |
$(3,5)(4,6)$ |
$-1$ |
$-1$ |
| $20$ |
$3$ |
$(1,4,2)(3,6,5)$ |
$0$ |
$0$ |
| $12$ |
$5$ |
$(1,4,5,6,2)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ |
$5$ |
$(1,5,2,4,6)$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
