Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 20 a + 19 + \left(15 a + 13\right)\cdot 41 + \left(35 a + 24\right)\cdot 41^{2} + \left(7 a + 16\right)\cdot 41^{3} + \left(25 a + 8\right)\cdot 41^{4} + \left(4 a + 9\right)\cdot 41^{5} + \left(7 a + 17\right)\cdot 41^{6} + \left(26 a + 28\right)\cdot 41^{7} + \left(4 a + 32\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 a + 39 + \left(33 a + 13\right)\cdot 41 + \left(11 a + 40\right)\cdot 41^{2} + \left(24 a + 30\right)\cdot 41^{3} + \left(5 a + 24\right)\cdot 41^{4} + \left(8 a + 32\right)\cdot 41^{5} + 19\cdot 41^{6} + \left(25 a + 22\right)\cdot 41^{7} + \left(14 a + 37\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 17\cdot 41 + 31\cdot 41^{2} + 34\cdot 41^{3} + 41^{4} + 30\cdot 41^{5} + 33\cdot 41^{6} + 7\cdot 41^{7} + 5\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 a + 31 + \left(7 a + 20\right)\cdot 41 + \left(29 a + 1\right)\cdot 41^{2} + \left(16 a + 10\right)\cdot 41^{3} + \left(35 a + 17\right)\cdot 41^{4} + \left(32 a + 10\right)\cdot 41^{5} + \left(40 a + 12\right)\cdot 41^{6} + \left(15 a + 15\right)\cdot 41^{7} + \left(26 a + 15\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 + 17\cdot 41 + 33\cdot 41^{2} + 25\cdot 41^{3} + 35\cdot 41^{4} + 41^{5} + 6\cdot 41^{6} + 31\cdot 41^{7} + 11\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 a + 38 + \left(25 a + 39\right)\cdot 41 + \left(5 a + 32\right)\cdot 41^{2} + \left(33 a + 4\right)\cdot 41^{3} + \left(15 a + 35\right)\cdot 41^{4} + \left(36 a + 38\right)\cdot 41^{5} + \left(33 a + 33\right)\cdot 41^{6} + \left(14 a + 17\right)\cdot 41^{7} + \left(36 a + 20\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(5,6)$ |
| $(1,6)$ |
| $(1,3,2)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,6)(2,4)(3,5)$ |
$-3$ |
| $3$ |
$2$ |
$(1,6)$ |
$1$ |
| $3$ |
$2$ |
$(1,6)(3,5)$ |
$-1$ |
| $6$ |
$2$ |
$(2,3)(4,5)$ |
$-1$ |
| $6$ |
$2$ |
$(1,6)(2,3)(4,5)$ |
$1$ |
| $8$ |
$3$ |
$(1,3,2)(4,6,5)$ |
$0$ |
| $6$ |
$4$ |
$(1,5,6,3)$ |
$-1$ |
| $6$ |
$4$ |
$(1,6)(2,5,4,3)$ |
$1$ |
| $8$ |
$6$ |
$(1,5,4,6,3,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
