Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 193 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 41 + 106\cdot 193 + 77\cdot 193^{2} + 107\cdot 193^{3} + 26\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 60 + 42\cdot 193 + 152\cdot 193^{2} + 53\cdot 193^{3} + 178\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 65 + 48\cdot 193 + 100\cdot 193^{2} + 191\cdot 193^{3} + 115\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 84 + 177\cdot 193 + 174\cdot 193^{2} + 137\cdot 193^{3} + 74\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 109 + 15\cdot 193 + 18\cdot 193^{2} + 55\cdot 193^{3} + 118\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 128 + 144\cdot 193 + 92\cdot 193^{2} + 193^{3} + 77\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 133 + 150\cdot 193 + 40\cdot 193^{2} + 139\cdot 193^{3} + 14\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 152 + 86\cdot 193 + 115\cdot 193^{2} + 85\cdot 193^{3} + 166\cdot 193^{4} +O\left(193^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,7)(3,6)(4,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,4,6)(2,5,3,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
