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:
| $r_{ 1 }$ |
$=$ |
$ 6 a + 24 + 16\cdot 41 + \left(20 a + 10\right)\cdot 41^{2} + \left(34 a + 7\right)\cdot 41^{3} + \left(30 a + 21\right)\cdot 41^{4} + \left(5 a + 3\right)\cdot 41^{5} + \left(22 a + 25\right)\cdot 41^{6} + \left(15 a + 25\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 a + \left(16 a + 32\right)\cdot 41 + \left(16 a + 20\right)\cdot 41^{2} + 38 a\cdot 41^{3} + \left(23 a + 17\right)\cdot 41^{4} + \left(7 a + 29\right)\cdot 41^{5} + \left(40 a + 36\right)\cdot 41^{6} + \left(16 a + 38\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 a + 10 + \left(24 a + 23\right)\cdot 41 + \left(24 a + 12\right)\cdot 41^{2} + \left(2 a + 17\right)\cdot 41^{3} + \left(17 a + 9\right)\cdot 41^{4} + \left(33 a + 28\right)\cdot 41^{5} + 26\cdot 41^{6} + \left(24 a + 8\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 a + 1 + \left(40 a + 11\right)\cdot 41 + \left(20 a + 29\right)\cdot 41^{2} + \left(6 a + 8\right)\cdot 41^{3} + \left(10 a + 38\right)\cdot 41^{4} + \left(35 a + 30\right)\cdot 41^{5} + \left(18 a + 3\right)\cdot 41^{6} + \left(25 a + 9\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 + 40\cdot 41 + 8\cdot 41^{2} + 7\cdot 41^{3} + 37\cdot 41^{4} + 30\cdot 41^{5} + 30\cdot 41^{6} + 40\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,5,4,3,2)$ |
| $(2,5)(3,4)$ |
| $(2,4,5,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $5$ |
$2$ |
$(1,4)(2,3)$ |
$0$ |
| $5$ |
$4$ |
$(1,3,4,2)$ |
$0$ |
| $5$ |
$4$ |
$(1,2,4,3)$ |
$0$ |
| $4$ |
$5$ |
$(1,5,4,3,2)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
