Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 44\cdot 83 + 21\cdot 83^{2} + 79\cdot 83^{3} + 30\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 69\cdot 83 + 50\cdot 83^{2} + 37\cdot 83^{3} + 5\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 + 22\cdot 83 + 13\cdot 83^{2} + 30\cdot 83^{3} + 37\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 42 + 71\cdot 83 + 76\cdot 83^{2} + 51\cdot 83^{3} + 53\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 43 + 47\cdot 83 + 42\cdot 83^{2} + 71\cdot 83^{3} + 11\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 46 + 42\cdot 83 + 8\cdot 83^{2} + 31\cdot 83^{3} + 82\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 71 + 31\cdot 83 + 10\cdot 83^{2} + 67\cdot 83^{3} + 40\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 75 + 2\cdot 83 + 25\cdot 83^{2} + 46\cdot 83^{3} + 69\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,5,8)(2,7,3,6)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,4)(3,8)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,4,5,8)(2,7,3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
