Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 90\cdot 139 + 24\cdot 139^{2} + 123\cdot 139^{3} + 123\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 + 16\cdot 139 + 19\cdot 139^{2} + 16\cdot 139^{3} + 56\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 40 + 60\cdot 139 + 101\cdot 139^{2} + 96\cdot 139^{3} + 94\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 52 + 43\cdot 139 + 56\cdot 139^{2} + 82\cdot 139^{3} + 129\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 55 + 23\cdot 139 + 64\cdot 139^{2} + 24\cdot 139^{3} + 88\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 64 + 90\cdot 139 + 36\cdot 139^{2} + 105\cdot 139^{3} + 117\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 78 + 59\cdot 139 + 103\cdot 139^{2} + 125\cdot 139^{3} + 128\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 111 + 33\cdot 139 + 11\cdot 139^{2} + 121\cdot 139^{3} + 94\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,8)(5,7)$ |
| $(1,4,2,8)(3,7,6,5)$ |
| $(1,3)(2,6)(4,5)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,6)(4,8)(5,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,6)(4,5)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,5)(3,4)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,2,8)(3,7,6,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
