Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 11 + 60\cdot 139 + 91\cdot 139^{2} + 92\cdot 139^{3} + 124\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 73\cdot 139 + 135\cdot 139^{2} + 56\cdot 139^{3} + 121\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 + 27\cdot 139 + 88\cdot 139^{2} + 32\cdot 139^{3} + 78\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 + 82\cdot 139 + 58\cdot 139^{2} + 97\cdot 139^{3} + 64\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 + 21\cdot 139 + 135\cdot 139^{2} + 29\cdot 139^{3} + 130\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 94 + 62\cdot 139 + 131\cdot 139^{2} + 30\cdot 139^{3} + 106\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 99 + 117\cdot 139 + 101\cdot 139^{2} + 95\cdot 139^{3} + 92\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 134 + 111\cdot 139 + 91\cdot 139^{2} + 119\cdot 139^{3} + 115\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,7)(5,8)$ |
| $(1,3,8,4)(2,7,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,5)(3,4)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,7)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,3)(4,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,3,8,4)(2,7,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
