Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 48\cdot 139 + 82\cdot 139^{2} + 93\cdot 139^{3} + 125\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 67\cdot 139 + 20\cdot 139^{2} + 74\cdot 139^{3} + 130\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 38\cdot 139 + 101\cdot 139^{2} + 72\cdot 139^{3} + 137\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 41 + 44\cdot 139 + 126\cdot 139^{2} + 137\cdot 139^{3} + 111\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 53 + 15\cdot 139 + 68\cdot 139^{2} + 136\cdot 139^{3} + 118\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 75 + 8\cdot 139 + 107\cdot 139^{2} + 112\cdot 139^{3} + 41\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 87 + 118\cdot 139 + 48\cdot 139^{2} + 111\cdot 139^{3} + 48\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 132 + 76\cdot 139 + 139^{2} + 95\cdot 139^{3} + 118\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,5,6)(3,4,7,8)$ |
| $(1,3)(2,8)(4,6)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,8)(4,6)(5,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,5,6)(3,4,7,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
