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:
| $r_{ 1 }$ |
$=$ |
$ 30 + 4\cdot 41 + 38\cdot 41^{2} + 18\cdot 41^{3} + 21\cdot 41^{4} + 13\cdot 41^{5} + 32\cdot 41^{6} + 22\cdot 41^{7} + 21\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ a + 17 + \left(32 a + 11\right)\cdot 41 + \left(39 a + 33\right)\cdot 41^{2} + \left(10 a + 8\right)\cdot 41^{3} + \left(35 a + 34\right)\cdot 41^{4} + \left(39 a + 24\right)\cdot 41^{5} + \left(13 a + 10\right)\cdot 41^{6} + \left(37 a + 22\right)\cdot 41^{7} + \left(17 a + 9\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a + 3 + \left(31 a + 8\right)\cdot 41 + \left(7 a + 8\right)\cdot 41^{2} + \left(38 a + 33\right)\cdot 41^{3} + \left(19 a + 28\right)\cdot 41^{4} + \left(6 a + 4\right)\cdot 41^{5} + \left(24 a + 1\right)\cdot 41^{6} + \left(14 a + 5\right)\cdot 41^{7} + \left(34 a + 18\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 a + 16 + \left(9 a + 2\right)\cdot 41 + 33 a\cdot 41^{2} + \left(2 a + 17\right)\cdot 41^{3} + \left(21 a + 9\right)\cdot 41^{4} + \left(34 a + 4\right)\cdot 41^{5} + \left(16 a + 26\right)\cdot 41^{6} + \left(26 a + 24\right)\cdot 41^{7} + \left(6 a + 24\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 a + 20 + \left(8 a + 24\right)\cdot 41 + \left(a + 38\right)\cdot 41^{2} + \left(30 a + 1\right)\cdot 41^{3} + \left(5 a + 6\right)\cdot 41^{4} + \left(a + 27\right)\cdot 41^{5} + \left(27 a + 12\right)\cdot 41^{6} + \left(3 a + 38\right)\cdot 41^{7} + \left(23 a + 25\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 38 + 30\cdot 41 + 4\cdot 41^{2} + 2\cdot 41^{3} + 23\cdot 41^{4} + 7\cdot 41^{5} + 40\cdot 41^{6} + 9\cdot 41^{7} + 23\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(4,6)$ |
| $(1,2,3)(4,6,5)$ |
| $(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(3,4)$ | $1$ |
| $3$ | $2$ | $(1,6)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,2)(5,6)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,3,6,4)$ | $1$ |
| $6$ | $4$ | $(1,5,6,2)(3,4)$ | $-1$ |
| $8$ | $6$ | $(1,2,3,6,5,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
