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:
| $r_{ 1 }$ |
$=$ |
$ 21 a + 36 + \left(20 a + 14\right)\cdot 41 + \left(11 a + 1\right)\cdot 41^{2} + \left(40 a + 1\right)\cdot 41^{3} + \left(3 a + 34\right)\cdot 41^{4} + \left(7 a + 1\right)\cdot 41^{5} + \left(6 a + 37\right)\cdot 41^{6} + \left(27 a + 15\right)\cdot 41^{7} + \left(17 a + 7\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 a + 9 + \left(14 a + 10\right)\cdot 41 + \left(11 a + 10\right)\cdot 41^{2} + \left(20 a + 37\right)\cdot 41^{3} + \left(8 a + 2\right)\cdot 41^{4} + \left(23 a + 25\right)\cdot 41^{5} + 39 a\cdot 41^{6} + \left(24 a + 4\right)\cdot 41^{7} + \left(34 a + 13\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 a + 17 + \left(20 a + 14\right)\cdot 41 + \left(29 a + 15\right)\cdot 41^{2} + 28\cdot 41^{3} + \left(37 a + 5\right)\cdot 41^{4} + \left(33 a + 19\right)\cdot 41^{5} + \left(34 a + 7\right)\cdot 41^{6} + \left(13 a + 9\right)\cdot 41^{7} + \left(23 a + 33\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 32 + 41 + 7\cdot 41^{2} + 35\cdot 41^{3} + 35\cdot 41^{4} + 3\cdot 41^{5} + 23\cdot 41^{6} + 12\cdot 41^{7} + 20\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 23 + 18\cdot 41^{2} + 16\cdot 41^{3} + 36\cdot 41^{4} + 27\cdot 41^{5} + 40\cdot 41^{6} + 39\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 a + 7 + \left(26 a + 40\right)\cdot 41 + \left(29 a + 29\right)\cdot 41^{2} + \left(20 a + 4\right)\cdot 41^{3} + \left(32 a + 8\right)\cdot 41^{4} + \left(17 a + 4\right)\cdot 41^{5} + \left(a + 14\right)\cdot 41^{6} + \left(16 a + 39\right)\cdot 41^{7} + \left(6 a + 9\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)$ |
| $(1,2)$ |
| $(1,4,3)(2,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-3$ |
| $3$ | $2$ | $(1,2)$ | $1$ |
| $3$ | $2$ | $(1,2)(4,5)$ | $-1$ |
| $6$ | $2$ | $(3,4)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,4,3)(2,5,6)$ | $0$ |
| $6$ | $4$ | $(1,5,2,4)$ | $-1$ |
| $6$ | $4$ | $(1,2)(3,5,6,4)$ | $1$ |
| $8$ | $6$ | $(1,5,6,2,4,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
