Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 19 a + \left(11 a + 18\right)\cdot 41 + \left(25 a + 27\right)\cdot 41^{2} + \left(26 a + 24\right)\cdot 41^{3} + \left(3 a + 29\right)\cdot 41^{4} + \left(15 a + 22\right)\cdot 41^{5} + \left(23 a + 20\right)\cdot 41^{6} + \left(36 a + 25\right)\cdot 41^{7} + \left(a + 15\right)\cdot 41^{8} + 38\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 a + 16 + \left(29 a + 33\right)\cdot 41 + \left(15 a + 9\right)\cdot 41^{2} + \left(14 a + 38\right)\cdot 41^{3} + \left(37 a + 13\right)\cdot 41^{4} + \left(25 a + 23\right)\cdot 41^{5} + \left(17 a + 34\right)\cdot 41^{6} + \left(4 a + 29\right)\cdot 41^{7} + \left(39 a + 25\right)\cdot 41^{8} + \left(40 a + 36\right)\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 + 6\cdot 41 + 31\cdot 41^{2} + 37\cdot 41^{3} + 35\cdot 41^{4} + 25\cdot 41^{5} + 41^{6} + 24\cdot 41^{7} + 6\cdot 41^{8} + 12\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 a + \left(29 a + 23\right)\cdot 41 + \left(15 a + 13\right)\cdot 41^{2} + \left(14 a + 16\right)\cdot 41^{3} + \left(37 a + 11\right)\cdot 41^{4} + \left(25 a + 18\right)\cdot 41^{5} + \left(17 a + 20\right)\cdot 41^{6} + \left(4 a + 15\right)\cdot 41^{7} + \left(39 a + 25\right)\cdot 41^{8} + \left(40 a + 2\right)\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 a + 25 + \left(11 a + 7\right)\cdot 41 + \left(25 a + 31\right)\cdot 41^{2} + \left(26 a + 2\right)\cdot 41^{3} + \left(3 a + 27\right)\cdot 41^{4} + \left(15 a + 17\right)\cdot 41^{5} + \left(23 a + 6\right)\cdot 41^{6} + \left(36 a + 11\right)\cdot 41^{7} + \left(a + 15\right)\cdot 41^{8} + 4\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 15 + 34\cdot 41 + 9\cdot 41^{2} + 3\cdot 41^{3} + 5\cdot 41^{4} + 15\cdot 41^{5} + 39\cdot 41^{6} + 16\cdot 41^{7} + 34\cdot 41^{8} + 28\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,5)$ |
| $(1,4)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(3,6)$ | $1$ |
| $3$ | $2$ | $(1,4)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(4,5)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,6)(4,5)$ | $1$ |
| $8$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(1,3,4,6)$ | $-1$ |
| $6$ | $4$ | $(1,4)(2,3,5,6)$ | $1$ |
| $8$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
