Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 + 59\cdot 139 + 57\cdot 139^{2} + 110\cdot 139^{3} + 72\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 29 + 123\cdot 139 + 82\cdot 139^{2} + 134\cdot 139^{3} + 85\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 42 + 22\cdot 139 + 10\cdot 139^{2} + 64\cdot 139^{3} + 92\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 66 + 128\cdot 139 + 63\cdot 139^{2} + 77\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 70 + 40\cdot 139 + 27\cdot 139^{2} + 60\cdot 139^{3} + 124\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 80 + 82\cdot 139 + 36\cdot 139^{2} + 34\cdot 139^{3} + 32\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 118 + 124\cdot 139 + 14\cdot 139^{2} + 66\cdot 139^{3} + 110\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 138 + 113\cdot 139 + 123\cdot 139^{2} + 85\cdot 139^{3} + 99\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3,8)(4,7,6,5)$ |
| $(1,4)(2,5)(3,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,8)(4,6)(5,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,4)(2,5)(3,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,3,8)(4,7,6,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
