Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 29 + 44\cdot 139 + 139^{2} + 120\cdot 139^{3} + 56\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 + 83\cdot 139 + 12\cdot 139^{2} + 127\cdot 139^{3} + 127\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 44 + 58\cdot 139 + 86\cdot 139^{2} + 2\cdot 139^{3} + 113\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 79 + 75\cdot 139 + 5\cdot 139^{2} + 60\cdot 139^{3} + 122\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 116 + 75\cdot 139 + 46\cdot 139^{2} + 49\cdot 139^{3} + 133\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 124 + 103\cdot 139 + 38\cdot 139^{2} + 111\cdot 139^{3} + 102\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 133 + 44\cdot 139 + 82\cdot 139^{2} + 15\cdot 139^{3} + 32\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 136 + 69\cdot 139 + 4\cdot 139^{2} + 70\cdot 139^{3} + 6\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,4)(3,8)(5,7)$ |
| $(1,7,6,8)(2,5,4,3)$ |
| $(1,2,6,4)(3,8,5,7)$ |
| $(1,6)(2,4)(3,5)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)(7,8)$ | $-2$ |
| $4$ | $2$ | $(2,4)(3,8)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,2,6,4)(3,8,5,7)$ | $0$ |
| $4$ | $4$ | $(1,7,6,8)(2,5,4,3)$ | $0$ |
| $2$ | $8$ | $(1,5,2,7,6,3,4,8)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,3,2,8,6,5,4,7)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
