Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 61\cdot 83 + 60\cdot 83^{2} + 33\cdot 83^{3} + 42\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 74\cdot 83 + 64\cdot 83^{2} + 12\cdot 83^{3} + 25\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 + 2\cdot 83 + 66\cdot 83^{2} + 39\cdot 83^{3} + 77\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 + 35\cdot 83 + 7\cdot 83^{2} + 31\cdot 83^{3} + 3\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 + 45\cdot 83 + 8\cdot 83^{2} + 14\cdot 83^{3} + 3\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 38 + 57\cdot 83 + 30\cdot 83^{2} + 78\cdot 83^{3} + 42\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 65 + 28\cdot 83 + 57\cdot 83^{2} + 79\cdot 83^{3} + 20\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 67 + 27\cdot 83 + 36\cdot 83^{2} + 42\cdot 83^{3} + 33\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,7)(4,6)(5,8)$ |
| $(1,4)(2,5)(3,8)(6,7)$ |
| $(1,3)(2,7)(4,8)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,7)(4,8)(5,6)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,7)(4,6)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,5)(3,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,5,3,6)(2,4,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
