Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 107 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 88\cdot 107 + 33\cdot 107^{2} + 38\cdot 107^{3} + 23\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 1 + 19\cdot 107 + 73\cdot 107^{2} + 68\cdot 107^{3} + 83\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 57\cdot 107 + 64\cdot 107^{2} + 51\cdot 107^{3} + 73\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 + 87\cdot 107 + 60\cdot 107^{2} + 39\cdot 107^{3} + 56\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 51 + 50\cdot 107 + 15\cdot 107^{2} + 54\cdot 107^{3} +O\left(107^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 57 + 56\cdot 107 + 91\cdot 107^{2} + 52\cdot 107^{3} + 106\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 69 + 19\cdot 107 + 46\cdot 107^{2} + 67\cdot 107^{3} + 50\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 90 + 49\cdot 107 + 42\cdot 107^{2} + 55\cdot 107^{3} + 33\cdot 107^{4} +O\left(107^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,7)(4,8)(5,6)$ |
| $(1,3)(2,8)(4,6)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,8,6,7)(2,3,5,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
