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:
| $r_{ 1 }$ |
$=$ |
$ 39 + 19\cdot 41 + 35\cdot 41^{2} + 5\cdot 41^{3} + 32\cdot 41^{4} + 9\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 a + 16 + \left(40 a + 36\right)\cdot 41 + \left(6 a + 16\right)\cdot 41^{2} + \left(9 a + 36\right)\cdot 41^{3} + \left(27 a + 9\right)\cdot 41^{4} + \left(27 a + 26\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 + 4\cdot 41 + 35\cdot 41^{2} + 37\cdot 41^{3} + 9\cdot 41^{4} + 3\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 a + 32 + \left(30 a + 36\right)\cdot 41 + \left(9 a + 39\right)\cdot 41^{2} + \left(36 a + 4\right)\cdot 41^{3} + \left(14 a + 31\right)\cdot 41^{4} + \left(a + 26\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 a + 24 + \left(10 a + 34\right)\cdot 41 + \left(31 a + 38\right)\cdot 41^{2} + \left(4 a + 21\right)\cdot 41^{3} + \left(26 a + 39\right)\cdot 41^{4} + \left(39 a + 15\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 a + 22 + 31\cdot 41 + \left(34 a + 38\right)\cdot 41^{2} + \left(31 a + 15\right)\cdot 41^{3} + 13 a\cdot 41^{4} + 13 a\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,4)(3,6,5)$ |
| $(1,3)$ |
| $(4,5)$ |
| $(2,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,3)(2,6)(4,5)$ |
$-3$ |
| $3$ |
$2$ |
$(1,3)$ |
$1$ |
| $3$ |
$2$ |
$(1,3)(2,6)$ |
$-1$ |
| $4$ |
$3$ |
$(1,2,4)(3,6,5)$ |
$0$ |
| $4$ |
$3$ |
$(1,4,2)(3,5,6)$ |
$0$ |
| $4$ |
$6$ |
$(1,6,5,3,2,4)$ |
$0$ |
| $4$ |
$6$ |
$(1,4,2,3,5,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
