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