Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 75\cdot 139 + 31\cdot 139^{2} + 43\cdot 139^{3} + 53\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 52\cdot 139 + 103\cdot 139^{2} + 15\cdot 139^{3} + 132\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 71\cdot 139 + 131\cdot 139^{2} + 95\cdot 139^{3} + 134\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 86 + 113\cdot 139 + 110\cdot 139^{2} + 39\cdot 139^{3} + 73\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 87 + 115\cdot 139 + 40\cdot 139^{2} + 34\cdot 139^{3} + 74\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 102 + 49\cdot 139 + 131\cdot 139^{2} + 115\cdot 139^{3} + 14\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 123 + 110\cdot 139 + 101\cdot 139^{2} + 39\cdot 139^{3} + 28\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 124 + 106\cdot 139 + 43\cdot 139^{2} + 32\cdot 139^{3} + 45\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,5,6)(3,4,7,8)$ |
| $(1,8,5,4)(2,7,6,3)$ |
| $(1,4,5,8)(2,7,6,3)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-2$ |
| $2$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ |
| $2$ | $2$ | $(1,5)(4,8)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ | $0$ |
| $1$ | $4$ | $(1,4,5,8)(2,7,6,3)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,8,5,4)(2,3,6,7)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,2,5,6)(3,4,7,8)$ | $0$ |
| $2$ | $4$ | $(1,8,5,4)(2,7,6,3)$ | $0$ |
| $2$ | $4$ | $(1,3,5,7)(2,8,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
