Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 139 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 92\cdot 139 + 27\cdot 139^{2} + 102\cdot 139^{3} + 139^{4} + 123\cdot 139^{5} +O\left(139^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 63\cdot 139 + 108\cdot 139^{2} + 100\cdot 139^{3} + 8\cdot 139^{4} + 134\cdot 139^{5} +O\left(139^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 + 37\cdot 139 + 16\cdot 139^{2} + 8\cdot 139^{3} + 124\cdot 139^{4} + 34\cdot 139^{5} +O\left(139^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 + 8\cdot 139 + 97\cdot 139^{2} + 6\cdot 139^{3} + 131\cdot 139^{4} + 45\cdot 139^{5} +O\left(139^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 103 + 130\cdot 139 + 41\cdot 139^{2} + 132\cdot 139^{3} + 7\cdot 139^{4} + 93\cdot 139^{5} +O\left(139^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 115 + 101\cdot 139 + 122\cdot 139^{2} + 130\cdot 139^{3} + 14\cdot 139^{4} + 104\cdot 139^{5} +O\left(139^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 118 + 75\cdot 139 + 30\cdot 139^{2} + 38\cdot 139^{3} + 130\cdot 139^{4} + 4\cdot 139^{5} +O\left(139^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 130 + 46\cdot 139 + 111\cdot 139^{2} + 36\cdot 139^{3} + 137\cdot 139^{4} + 15\cdot 139^{5} +O\left(139^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,4,7,6)(2,3,8,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
