Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 43\cdot 47 + 15\cdot 47^{2} + 16\cdot 47^{3} + 16\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 7\cdot 47 + 47^{2} + 36\cdot 47^{3} + 9\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 35\cdot 47 + 32\cdot 47^{2} + 43\cdot 47^{3} + 25\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 + 5\cdot 47 + 33\cdot 47^{2} + 4\cdot 47^{3} + 41\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 + 32\cdot 47 + 2\cdot 47^{2} + 20\cdot 47^{3} + 19\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 31 + 44\cdot 47 + 14\cdot 47^{2} + 40\cdot 47^{3} + 4\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 39 + 13\cdot 47 + 29\cdot 47^{2} + 44\cdot 47^{3} + 43\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 45 + 5\cdot 47 + 11\cdot 47^{2} + 29\cdot 47^{3} + 26\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,8,6,7,3,5,4)$ |
| $(1,5,7,8)(2,4,3,6)$ |
| $(1,7)(2,3)(4,6)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,7)(2,3)(4,6)(5,8)$ | $-1$ |
| $1$ | $4$ | $(1,8,7,5)(2,6,3,4)$ | $-\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,5,7,8)(2,4,3,6)$ | $\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,2,8,6,7,3,5,4)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,6,5,2,7,4,8,3)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,3,8,4,7,2,5,6)$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,4,5,3,7,6,8,2)$ | $\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
