Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 38\cdot 47 + 33\cdot 47^{2} + 46\cdot 47^{3} + 25\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 8\cdot 47 + 6\cdot 47^{2} + 36\cdot 47^{3} + 37\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 + 29\cdot 47 + 27\cdot 47^{2} + 37\cdot 47^{3} + 39\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 + 39\cdot 47 + 12\cdot 47^{2} + 17\cdot 47^{3} + 37\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 + 47 + 35\cdot 47^{3} + 8\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 + 24\cdot 47 + 10\cdot 47^{2} + 45\cdot 47^{3} + 37\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 34 + 12\cdot 47 + 47^{2} + 32\cdot 47^{3} + 45\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 44 + 33\cdot 47 + 47^{2} + 32\cdot 47^{3} + 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,6)(4,7)(5,8)$ |
| $(1,4,6,5)(2,7,3,8)$ |
| $(1,2)(3,6)(4,8)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,3)(4,5)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,8)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,4)(3,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,4,6,5)(2,7,3,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
