Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 5\cdot 83 + 14\cdot 83^{2} + 25\cdot 83^{3} + 41\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 + 53\cdot 83 + 50\cdot 83^{2} + 55\cdot 83^{3} + 61\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 37 + 76\cdot 83 + 48\cdot 83^{2} + 69\cdot 83^{3} + 71\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 60 + 57\cdot 83 + 77\cdot 83^{2} + 82\cdot 83^{3} + 43\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 65 + 26\cdot 83 + 25\cdot 83^{2} + 71\cdot 83^{3} + 8\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 69 + 80\cdot 83 + 75\cdot 83^{2} + 13\cdot 83^{3} + 54\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 74 + 49\cdot 83 + 23\cdot 83^{2} + 2\cdot 83^{3} + 19\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 80 + 64\cdot 83 + 15\cdot 83^{2} + 11\cdot 83^{3} + 31\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,7,4)(3,5,6,8)$ |
| $(1,3)(2,8)(4,5)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,4)(3,6)(5,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,8)(4,5)(6,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,8)(2,6)(3,4)(5,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,7,4)(3,5,6,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
