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:
| $r_{ 1 }$ |
$=$ |
$ 11 + 22\cdot 41 + 41^{2} + 34\cdot 41^{3} + 40\cdot 41^{4} + 15\cdot 41^{5} + 30\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 + 7\cdot 41 + 12\cdot 41^{2} + 21\cdot 41^{3} + 24\cdot 41^{4} + 19\cdot 41^{5} + 35\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 a + 4 + \left(23 a + 2\right)\cdot 41 + \left(a + 28\right)\cdot 41^{2} + \left(33 a + 34\right)\cdot 41^{3} + \left(a + 29\right)\cdot 41^{4} + \left(9 a + 27\right)\cdot 41^{5} + \left(30 a + 40\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 a + 1 + \left(40 a + 9\right)\cdot 41 + \left(15 a + 32\right)\cdot 41^{2} + \left(14 a + 18\right)\cdot 41^{3} + \left(32 a + 12\right)\cdot 41^{4} + \left(23 a + 4\right)\cdot 41^{5} + \left(16 a + 16\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 a + 37 + 35\cdot 41 + \left(25 a + 39\right)\cdot 41^{2} + \left(26 a + 4\right)\cdot 41^{3} + \left(8 a + 13\right)\cdot 41^{4} + \left(17 a + 2\right)\cdot 41^{5} + \left(24 a + 1\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 a + 3 + \left(17 a + 5\right)\cdot 41 + \left(39 a + 9\right)\cdot 41^{2} + \left(7 a + 9\right)\cdot 41^{3} + \left(39 a + 2\right)\cdot 41^{4} + \left(31 a + 12\right)\cdot 41^{5} + \left(10 a + 40\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)$ |
| $(1,5)(2,4)$ |
| $(1,5,6)(2,4,3)$ |
| $(1,4,3)(2,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(4,5)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,5)$ | $-1$ |
| $8$ | $3$ | $(1,6,5)(2,3,4)$ | $0$ |
| $6$ | $4$ | $(1,2)(3,5,6,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
