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:
| $r_{ 1 }$ |
$=$ |
$ 5 a + 20 + \left(28 a + 15\right)\cdot 41 + \left(23 a + 39\right)\cdot 41^{2} + \left(9 a + 24\right)\cdot 41^{3} + \left(28 a + 21\right)\cdot 41^{4} + \left(16 a + 38\right)\cdot 41^{5} + \left(36 a + 11\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 36 a + 35 + \left(12 a + 12\right)\cdot 41 + 17 a\cdot 41^{2} + \left(31 a + 30\right)\cdot 41^{3} + \left(12 a + 14\right)\cdot 41^{4} + \left(24 a + 19\right)\cdot 41^{5} + \left(4 a + 22\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 2 + 10\cdot 41 + 5\cdot 41^{2} + 15\cdot 41^{3} + 14\cdot 41^{4} + 22\cdot 41^{5} + 15\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 + 5\cdot 41 + 7\cdot 41^{2} + 14\cdot 41^{3} + 37\cdot 41^{4} + 20\cdot 41^{5} + 33\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 a + \left(20 a + 19\right)\cdot 41 + \left(21 a + 13\right)\cdot 41^{2} + \left(31 a + 3\right)\cdot 41^{3} + \left(13 a + 33\right)\cdot 41^{4} + \left(34 a + 27\right)\cdot 41^{5} + \left(40 a + 16\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 a + 19 + \left(20 a + 19\right)\cdot 41 + \left(19 a + 16\right)\cdot 41^{2} + \left(9 a + 35\right)\cdot 41^{3} + \left(27 a + 1\right)\cdot 41^{4} + \left(6 a + 35\right)\cdot 41^{5} + 22\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3)(4,6)$ |
| $(1,2)(5,6)$ |
| $(1,4)(2,6)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-2$ |
| $3$ | $2$ | $(1,2)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,4)(2,6)(3,5)$ | $0$ |
| $2$ | $3$ | $(1,3,2)(4,6,5)$ | $-1$ |
| $2$ | $6$ | $(1,6,3,5,2,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
