Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 12\cdot 139 + 33\cdot 139^{2} + 78\cdot 139^{3} + 24\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 + 107\cdot 139 + 127\cdot 139^{2} + 112\cdot 139^{3} + 44\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 + 91\cdot 139 + 12\cdot 139^{2} + 31\cdot 139^{3} + 6\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 41 + 18\cdot 139 + 49\cdot 139^{2} + 64\cdot 139^{3} + 33\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 98 + 120\cdot 139 + 89\cdot 139^{2} + 74\cdot 139^{3} + 105\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 115 + 47\cdot 139 + 126\cdot 139^{2} + 107\cdot 139^{3} + 132\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 124 + 31\cdot 139 + 11\cdot 139^{2} + 26\cdot 139^{3} + 94\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 129 + 126\cdot 139 + 105\cdot 139^{2} + 60\cdot 139^{3} + 114\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,3,7,5)(2,4,8,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,3,7,5)(2,4,8,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
