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:
| $r_{ 1 }$ |
$=$ |
$ 34 + 14\cdot 41 + 24\cdot 41^{2} + 19\cdot 41^{3} + 6\cdot 41^{4} + 41^{5} + 12\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 a + 17 + \left(39 a + 23\right)\cdot 41 + \left(33 a + 37\right)\cdot 41^{2} + \left(8 a + 29\right)\cdot 41^{3} + \left(17 a + 24\right)\cdot 41^{4} + \left(28 a + 40\right)\cdot 41^{5} + \left(31 a + 39\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 a + 30 + \left(27 a + 14\right)\cdot 41 + \left(2 a + 24\right)\cdot 41^{2} + \left(17 a + 38\right)\cdot 41^{3} + \left(28 a + 27\right)\cdot 41^{4} + \left(33 a + 31\right)\cdot 41^{5} + \left(15 a + 23\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 + 15\cdot 41 + 13\cdot 41^{2} + 7\cdot 41^{3} + 23\cdot 41^{4} + 26\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 37 a + 1 + \left(13 a + 10\right)\cdot 41 + \left(38 a + 5\right)\cdot 41^{2} + \left(23 a + 5\right)\cdot 41^{3} + \left(12 a + 14\right)\cdot 41^{4} + \left(7 a + 22\right)\cdot 41^{5} + \left(25 a + 37\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 26 a + 21 + \left(a + 3\right)\cdot 41 + \left(7 a + 18\right)\cdot 41^{2} + \left(32 a + 22\right)\cdot 41^{3} + \left(23 a + 26\right)\cdot 41^{4} + \left(12 a + 26\right)\cdot 41^{5} + \left(9 a + 24\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)$ |
| $(2,6)$ |
| $(1,3,2)(4,5,6)$ |
| $(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,6)(3,5)$ | $-3$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $3$ | $2$ | $(1,4)(2,6)$ | $-1$ |
| $4$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
| $4$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $4$ | $6$ | $(1,5,6,4,3,2)$ | $0$ |
| $4$ | $6$ | $(1,2,3,4,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
