Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 40\cdot 41 + 12\cdot 41^{2} + 37\cdot 41^{3} + 34\cdot 41^{4} + 36\cdot 41^{5} + 4\cdot 41^{6} + 8\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 a + 4 + \left(32 a + 9\right)\cdot 41 + \left(38 a + 14\right)\cdot 41^{2} + \left(30 a + 38\right)\cdot 41^{3} + \left(20 a + 28\right)\cdot 41^{4} + \left(33 a + 31\right)\cdot 41^{5} + \left(34 a + 28\right)\cdot 41^{6} + \left(26 a + 26\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 32 a + 34 + \left(12 a + 17\right)\cdot 41 + \left(35 a + 8\right)\cdot 41^{2} + \left(28 a + 17\right)\cdot 41^{3} + \left(10 a + 1\right)\cdot 41^{4} + \left(2 a + 4\right)\cdot 41^{5} + \left(37 a + 25\right)\cdot 41^{6} + \left(6 a + 24\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 a + 7 + \left(28 a + 24\right)\cdot 41 + \left(5 a + 19\right)\cdot 41^{2} + \left(12 a + 27\right)\cdot 41^{3} + \left(30 a + 4\right)\cdot 41^{4} + 38 a\cdot 41^{5} + \left(3 a + 11\right)\cdot 41^{6} + \left(34 a + 8\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 + 27\cdot 41 + 10\cdot 41^{2} + 33\cdot 41^{3} + 33\cdot 41^{4} + 20\cdot 41^{5} + 35\cdot 41^{6} + 23\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 a + 23 + \left(8 a + 4\right)\cdot 41 + \left(2 a + 16\right)\cdot 41^{2} + \left(10 a + 10\right)\cdot 41^{3} + \left(20 a + 19\right)\cdot 41^{4} + \left(7 a + 29\right)\cdot 41^{5} + \left(6 a + 17\right)\cdot 41^{6} + \left(14 a + 31\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,6)(3,4)$ |
| $(1,2,3,5,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,4)(3,6)$ |
$-2$ |
| $3$ |
$2$ |
$(2,6)(3,4)$ |
$0$ |
| $3$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$0$ |
| $2$ |
$3$ |
$(1,3,4)(2,5,6)$ |
$-1$ |
| $2$ |
$6$ |
$(1,2,3,5,4,6)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
