Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 7\cdot 139 + 19\cdot 139^{2} + 56\cdot 139^{3} + 131\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 + 73\cdot 139 + 25\cdot 139^{2} + 93\cdot 139^{3} + 78\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 + 53\cdot 139 + 93\cdot 139^{3} + 93\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 69 + 30\cdot 139 + 101\cdot 139^{2} + 94\cdot 139^{3} + 2\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 95 + 87\cdot 139 + 76\cdot 139^{2} + 116\cdot 139^{3} + 56\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 101 + 19\cdot 139 + 7\cdot 139^{2} + 85\cdot 139^{3} + 113\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 117 + 131\cdot 139 + 83\cdot 139^{2} + 53\cdot 139^{3} + 10\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 123 + 13\cdot 139 + 103\cdot 139^{2} + 102\cdot 139^{3} + 68\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,8)(3,5)(6,7)$ |
| $(1,6,4,7)(2,5,8,3)$ |
| $(1,2,4,8)(3,7,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,8)(3,5)(6,7)$ | $-2$ |
| $2$ | $4$ | $(1,2,4,8)(3,7,5,6)$ | $0$ |
| $2$ | $4$ | $(1,6,4,7)(2,5,8,3)$ | $0$ |
| $2$ | $4$ | $(1,5,4,3)(2,7,8,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
