Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 19 a + 33 + \left(25 a + 23\right)\cdot 41 + \left(22 a + 29\right)\cdot 41^{2} + \left(25 a + 35\right)\cdot 41^{3} + \left(21 a + 2\right)\cdot 41^{4} + \left(17 a + 17\right)\cdot 41^{5} + \left(27 a + 11\right)\cdot 41^{6} + \left(31 a + 33\right)\cdot 41^{7} + \left(25 a + 40\right)\cdot 41^{8} + \left(32 a + 5\right)\cdot 41^{9} + \left(33 a + 18\right)\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 17 + \left(23 a + 2\right)\cdot 41 + 21 a\cdot 41^{2} + \left(a + 34\right)\cdot 41^{3} + \left(8 a + 32\right)\cdot 41^{4} + \left(38 a + 26\right)\cdot 41^{5} + \left(3 a + 40\right)\cdot 41^{6} + \left(26 a + 6\right)\cdot 41^{7} + \left(30 a + 8\right)\cdot 41^{8} + \left(22 a + 30\right)\cdot 41^{9} + \left(25 a + 17\right)\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 29 a + 12 + \left(17 a + 19\right)\cdot 41 + 19 a\cdot 41^{2} + \left(39 a + 17\right)\cdot 41^{3} + \left(32 a + 14\right)\cdot 41^{4} + \left(2 a + 10\right)\cdot 41^{5} + \left(37 a + 14\right)\cdot 41^{6} + \left(14 a + 40\right)\cdot 41^{7} + \left(10 a + 32\right)\cdot 41^{8} + \left(18 a + 26\right)\cdot 41^{9} + \left(15 a + 30\right)\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 a + 8 + \left(15 a + 40\right)\cdot 41 + \left(18 a + 30\right)\cdot 41^{2} + \left(15 a + 7\right)\cdot 41^{3} + \left(19 a + 1\right)\cdot 41^{4} + \left(23 a + 7\right)\cdot 41^{5} + \left(13 a + 35\right)\cdot 41^{6} + \left(9 a + 18\right)\cdot 41^{7} + \left(15 a + 4\right)\cdot 41^{8} + \left(8 a + 37\right)\cdot 41^{9} + \left(7 a + 4\right)\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 37 + 40\cdot 41 + 32\cdot 41^{2} + 24\cdot 41^{3} + 29\cdot 41^{4} + 39\cdot 41^{5} + 24\cdot 41^{6} + 18\cdot 41^{8} + 38\cdot 41^{9} + 24\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 + 37\cdot 41 + 28\cdot 41^{2} + 3\cdot 41^{3} + 41^{4} + 22\cdot 41^{5} + 37\cdot 41^{6} + 22\cdot 41^{7} + 18\cdot 41^{8} + 25\cdot 41^{9} + 26\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(5,6)$ |
| $(1,5,2)(3,6,4)$ |
| $(2,5)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$-3$ |
| $3$ |
$2$ |
$(1,3)$ |
$1$ |
| $3$ |
$2$ |
$(1,3)(5,6)$ |
$-1$ |
| $6$ |
$2$ |
$(2,5)(4,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,3)(2,5)(4,6)$ |
$-1$ |
| $8$ |
$3$ |
$(1,5,2)(3,6,4)$ |
$0$ |
| $6$ |
$4$ |
$(1,6,3,5)$ |
$1$ |
| $6$ |
$4$ |
$(1,4,3,2)(5,6)$ |
$-1$ |
| $8$ |
$6$ |
$(1,6,4,3,5,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
