Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 139 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 103\cdot 139 + 78\cdot 139^{2} + 67\cdot 139^{3} + 125\cdot 139^{4} + 116\cdot 139^{5} +O\left(139^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 16\cdot 139 + 126\cdot 139^{2} + 121\cdot 139^{3} + 51\cdot 139^{4} + 34\cdot 139^{5} +O\left(139^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 31\cdot 139 + 32\cdot 139^{2} + 17\cdot 139^{3} + 4\cdot 139^{4} + 108\cdot 139^{5} +O\left(139^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 + 41\cdot 139 + 86\cdot 139^{2} + 37\cdot 139^{3} + 65\cdot 139^{4} + 18\cdot 139^{5} +O\left(139^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 100 + 97\cdot 139 + 52\cdot 139^{2} + 101\cdot 139^{3} + 73\cdot 139^{4} + 120\cdot 139^{5} +O\left(139^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 119 + 107\cdot 139 + 106\cdot 139^{2} + 121\cdot 139^{3} + 134\cdot 139^{4} + 30\cdot 139^{5} +O\left(139^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 130 + 122\cdot 139 + 12\cdot 139^{2} + 17\cdot 139^{3} + 87\cdot 139^{4} + 104\cdot 139^{5} +O\left(139^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 133 + 35\cdot 139 + 60\cdot 139^{2} + 71\cdot 139^{3} + 13\cdot 139^{4} + 22\cdot 139^{5} +O\left(139^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,8,7)(3,5,6,4)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
