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:
| $r_{ 1 }$ |
$=$ |
$ 28 a + 4 + \left(15 a + 5\right)\cdot 41 + \left(29 a + 38\right)\cdot 41^{2} + \left(7 a + 11\right)\cdot 41^{3} + \left(15 a + 5\right)\cdot 41^{4} + \left(36 a + 2\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 39 + 16\cdot 41 + 15\cdot 41^{2} + 37\cdot 41^{3} + 35\cdot 41^{4} + 13\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 14\cdot 41 + 11\cdot 41^{2} + 35\cdot 41^{3} + 23\cdot 41^{4} + 26\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 a + 6 + \left(25 a + 24\right)\cdot 41 + \left(11 a + 28\right)\cdot 41^{2} + \left(33 a + 5\right)\cdot 41^{3} + \left(25 a + 2\right)\cdot 41^{4} + \left(4 a + 14\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 a + 5 + \left(6 a + 36\right)\cdot 41 + \left(40 a + 39\right)\cdot 41^{2} + \left(36 a + 21\right)\cdot 41^{3} + \left(9 a + 31\right)\cdot 41^{4} + \left(7 a + 6\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 11 a + 13 + \left(34 a + 26\right)\cdot 41 + 30\cdot 41^{2} + \left(4 a + 10\right)\cdot 41^{3} + \left(31 a + 24\right)\cdot 41^{4} + \left(33 a + 18\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,4)$ |
| $(1,5,3,6,4,2)$ |
| $(1,4)(2,5)$ |
| $(2,6)(3,4)$ |
| $(1,3)(2,5)$ |
| $(3,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $3$ |
$2$ |
$(1,6)(2,3)(4,5)$ |
$0$ |
| $3$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$0$ |
| $9$ |
$2$ |
$(1,3)(2,5)$ |
$0$ |
| $2$ |
$3$ |
$(1,3,4)(2,5,6)$ |
$-2$ |
| $2$ |
$3$ |
$(1,4,3)(2,5,6)$ |
$-2$ |
| $4$ |
$3$ |
$(1,3,4)$ |
$1$ |
| $6$ |
$6$ |
$(1,5,3,6,4,2)$ |
$0$ |
| $6$ |
$6$ |
$(1,6,4,2,3,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
