Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 46\cdot 47 + 8\cdot 47^{2} + 15\cdot 47^{3} + 13\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 40\cdot 47 + 29\cdot 47^{2} + 25\cdot 47^{3} + 41\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 43\cdot 47 + 3\cdot 47^{2} + 23\cdot 47^{3} + 44\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 39\cdot 47 + 4\cdot 47^{2} + 34\cdot 47^{3} + 32\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 + 7\cdot 47 + 6\cdot 47^{2} + 22\cdot 47^{3} + 20\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 + 16\cdot 47 + 15\cdot 47^{2} + 35\cdot 47^{3} + 18\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 37 + 34\cdot 47 + 27\cdot 47^{2} + 10\cdot 47^{3} + 15\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 38 + 6\cdot 47 + 44\cdot 47^{2} + 21\cdot 47^{3} + 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,4,2)(3,5,8,7)$ |
| $(1,3,2,7,4,8,6,5)$ |
| $(1,4)(2,6)(3,8)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,6)(3,8)(5,7)$ | $-1$ |
| $1$ | $4$ | $(1,2,4,6)(3,7,8,5)$ | $-\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,6,4,2)(3,5,8,7)$ | $\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,3,2,7,4,8,6,5)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,7,6,3,4,5,2,8)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,8,2,5,4,3,6,7)$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,5,6,8,4,7,2,3)$ | $\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
