Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 45 + 26\cdot 139 + 131\cdot 139^{2} + 3\cdot 139^{3} + 115\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 54 + 39\cdot 139 + 54\cdot 139^{2} + 107\cdot 139^{3} + 80\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 61 + 81\cdot 139 + 117\cdot 139^{2} + 18\cdot 139^{3} + 7\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 119 + 130\cdot 139 + 113\cdot 139^{2} + 8\cdot 139^{3} + 75\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2)(3,4)$ |
| $(2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,4)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,3,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
