Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 20\cdot 83 + 65\cdot 83^{2} + 35\cdot 83^{3} + 44\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 + 15\cdot 83 + 16\cdot 83^{2} + 10\cdot 83^{3} + 68\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 46 + 50\cdot 83 + 66\cdot 83^{2} + 20\cdot 83^{3} + 82\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 49 + 19\cdot 83 + 76\cdot 83^{2} + 7\cdot 83^{3} + 9\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 66 + 47\cdot 83 + 50\cdot 83^{2} + 80\cdot 83^{3} + 72\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 71 + 82\cdot 83 + 33\cdot 83^{2} + 39\cdot 83^{3} + 63\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 74 + 40\cdot 83 + 82\cdot 83^{2} + 15\cdot 83^{3} +O\left(83^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 80 + 54\cdot 83 + 23\cdot 83^{2} + 38\cdot 83^{3} + 74\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,8)(4,5)(6,7)$ |
| $(1,2,6,5)(3,4,7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,7)(4,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,8)(4,5)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,7)(3,5)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,2,6,5)(3,4,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
