Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 83 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 78\cdot 83 + 19\cdot 83^{2} + 47\cdot 83^{3} + 83^{4} + 64\cdot 83^{5} +O\left(83^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 + 37\cdot 83 + 60\cdot 83^{2} + 67\cdot 83^{3} + 17\cdot 83^{4} + 80\cdot 83^{5} +O\left(83^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 24\cdot 83 + 67\cdot 83^{2} + 8\cdot 83^{3} + 64\cdot 83^{4} + 43\cdot 83^{5} +O\left(83^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 41 + 56\cdot 83 + 64\cdot 83^{2} + 40\cdot 83^{3} + 22\cdot 83^{5} +O\left(83^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 + 26\cdot 83 + 18\cdot 83^{2} + 42\cdot 83^{3} + 82\cdot 83^{4} + 60\cdot 83^{5} +O\left(83^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 60 + 58\cdot 83 + 15\cdot 83^{2} + 74\cdot 83^{3} + 18\cdot 83^{4} + 39\cdot 83^{5} +O\left(83^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 68 + 45\cdot 83 + 22\cdot 83^{2} + 15\cdot 83^{3} + 65\cdot 83^{4} + 2\cdot 83^{5} +O\left(83^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 80 + 4\cdot 83 + 63\cdot 83^{2} + 35\cdot 83^{3} + 81\cdot 83^{4} + 18\cdot 83^{5} +O\left(83^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3,5)(4,8,7,6)$ |
| $(1,4)(2,6)(3,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,7)(3,8)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,2,3,5)(4,8,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
