Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 21 + 112\cdot 139 + 86\cdot 139^{2} + 14\cdot 139^{3} + 74\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 54 + 106\cdot 139 + 59\cdot 139^{2} + 43\cdot 139^{3} + 110\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 66 + 91\cdot 139 + 9\cdot 139^{2} + 13\cdot 139^{3} + 120\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 67 + 7\cdot 139 + 136\cdot 139^{2} + 90\cdot 139^{3} + 132\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 72 + 131\cdot 139 + 2\cdot 139^{2} + 48\cdot 139^{3} + 6\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 73 + 47\cdot 139 + 129\cdot 139^{2} + 125\cdot 139^{3} + 18\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 85 + 32\cdot 139 + 79\cdot 139^{2} + 95\cdot 139^{3} + 28\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 118 + 26\cdot 139 + 52\cdot 139^{2} + 124\cdot 139^{3} + 64\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
