Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 59\cdot 83 + 28\cdot 83^{2} + 3\cdot 83^{3} + 25\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 42\cdot 83 + 59\cdot 83^{2} + 68\cdot 83^{3} + 73\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 + 77\cdot 83 + 49\cdot 83^{2} + 26\cdot 83^{3} + 17\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 + 54\cdot 83 + 41\cdot 83^{2} + 4\cdot 83^{3} + 67\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 + 28\cdot 83 + 41\cdot 83^{2} + 78\cdot 83^{3} + 15\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 59 + 5\cdot 83 + 33\cdot 83^{2} + 56\cdot 83^{3} + 65\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 69 + 40\cdot 83 + 23\cdot 83^{2} + 14\cdot 83^{3} + 9\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 77 + 23\cdot 83 + 54\cdot 83^{2} + 79\cdot 83^{3} + 57\cdot 83^{4} +O\left(83^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,7,4)(2,5,8,6)$ |
| $(1,2)(3,6)(4,5)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,8)(3,4)(5,6)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,6)(4,5)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,3)(4,8)(6,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,7,4)(2,5,8,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
