Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 a + 19 + \left(19 a + 3\right)\cdot 41 + \left(28 a + 6\right)\cdot 41^{2} + \left(4 a + 35\right)\cdot 41^{3} + \left(36 a + 26\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 a + 32 + \left(25 a + 6\right)\cdot 41 + \left(30 a + 39\right)\cdot 41^{2} + \left(18 a + 20\right)\cdot 41^{3} + \left(3 a + 14\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 8\cdot 41 + 23\cdot 41^{2} + 36\cdot 41^{3} + 34\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 + 3\cdot 41 + 4\cdot 41^{3} + 33\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 a + 8 + \left(21 a + 10\right)\cdot 41 + \left(12 a + 31\right)\cdot 41^{2} + \left(36 a + 20\right)\cdot 41^{3} + \left(4 a + 7\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 a + 14 + \left(15 a + 8\right)\cdot 41 + \left(10 a + 23\right)\cdot 41^{2} + \left(22 a + 5\right)\cdot 41^{3} + \left(37 a + 6\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,5)$ |
| $(1,2)(3,4)(5,6)$ |
| $(1,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,5)$ | $2$ |
| $9$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,3,5)$ | $1$ |
| $4$ | $3$ | $(1,3,5)(2,4,6)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,4,3,6,5,2)$ | $0$ |
| $12$ | $6$ | $(2,4,6)(3,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
