Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 20 a + 21 + \left(19 a + 9\right)\cdot 41 + \left(16 a + 18\right)\cdot 41^{2} + 13\cdot 41^{3} + \left(33 a + 3\right)\cdot 41^{4} + \left(15 a + 23\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 a + 19 + \left(33 a + 33\right)\cdot 41 + \left(3 a + 26\right)\cdot 41^{2} + \left(17 a + 34\right)\cdot 41^{3} + \left(13 a + 34\right)\cdot 41^{4} + \left(25 a + 39\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 + 18\cdot 41^{2} + 15\cdot 41^{3} + 39\cdot 41^{4} + 35\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 + 4\cdot 41 + 7\cdot 41^{2} + 20\cdot 41^{3} + 8\cdot 41^{4} + 7\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 a + 12 + \left(7 a + 27\right)\cdot 41 + \left(37 a + 4\right)\cdot 41^{2} + 23 a\cdot 41^{3} + \left(27 a + 17\right)\cdot 41^{4} + \left(15 a + 20\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 a + 40 + \left(21 a + 6\right)\cdot 41 + \left(24 a + 7\right)\cdot 41^{2} + \left(40 a + 39\right)\cdot 41^{3} + \left(7 a + 19\right)\cdot 41^{4} + \left(25 a + 37\right)\cdot 41^{5} +O\left(41^{ 6 }\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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(1,6)(3,4)$ | $-1$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $6$ | $2$ | $(1,3)(4,6)$ | $1$ |
| $6$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,4,6,3)$ | $1$ |
| $6$ | $4$ | $(1,4,6,3)(2,5)$ | $-1$ |
| $8$ | $6$ | $(1,4,5,6,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
