Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 29 + 10\cdot 139 + 134\cdot 139^{2} + 64\cdot 139^{3} + 7\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 41 + 120\cdot 139 + 75\cdot 139^{2} + 63\cdot 139^{3} + 14\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 59 + 29\cdot 139 + 41\cdot 139^{2} + 31\cdot 139^{3} + 76\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 69 + 138\cdot 139 + 16\cdot 139^{2} + 109\cdot 139^{3} + 55\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 71 + 122\cdot 139^{2} + 29\cdot 139^{3} + 83\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 81 + 109\cdot 139 + 97\cdot 139^{2} + 107\cdot 139^{3} + 62\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 99 + 18\cdot 139 + 63\cdot 139^{2} + 75\cdot 139^{3} + 124\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 111 + 128\cdot 139 + 4\cdot 139^{2} + 74\cdot 139^{3} + 131\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,5,7,6)(2,3,8,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
