Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 54\cdot 83 + 25\cdot 83^{2} + 55\cdot 83^{3} + 42\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 + 17\cdot 83 + 22\cdot 83^{2} + 63\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 + 52\cdot 83 + 45\cdot 83^{2} + 32\cdot 83^{3} + 25\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 + 67\cdot 83 + 40\cdot 83^{2} + 22\cdot 83^{3} + 37\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 44 + 15\cdot 83 + 42\cdot 83^{2} + 60\cdot 83^{3} + 45\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 57 + 30\cdot 83 + 37\cdot 83^{2} + 50\cdot 83^{3} + 57\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 64 + 65\cdot 83 + 60\cdot 83^{2} + 82\cdot 83^{3} + 19\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 82 + 28\cdot 83 + 57\cdot 83^{2} + 27\cdot 83^{3} + 40\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,8)(3,4)(5,6)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,6)(4,5)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,5,7,6)(2,3,8,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
