Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 24 + 37\cdot 139 + 53\cdot 139^{2} + 121\cdot 139^{3} + 92\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 36 + 128\cdot 139 + 70\cdot 139^{2} + 29\cdot 139^{3} + 21\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 42 + 129\cdot 139 + 35\cdot 139^{2} + 90\cdot 139^{3} + 121\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 46 + 79\cdot 139 + 132\cdot 139^{2} + 22\cdot 139^{3} + 39\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 74 + 61\cdot 139 + 4\cdot 139^{2} + 19\cdot 139^{3} + 135\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 105 + 28\cdot 139 + 112\cdot 139^{2} + 94\cdot 139^{3} + 61\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 110 + 18\cdot 139 + 131\cdot 139^{2} + 33\cdot 139^{3} + 128\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 122 + 72\cdot 139 + 15\cdot 139^{2} + 5\cdot 139^{3} + 95\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,4,5)(2,6,7,8)$ |
| $(1,2)(4,7)(6,8)$ |
| $(1,4)(2,7)(3,5)(6,8)$ |
| $(1,7,4,2)(3,6,5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,7)(3,5)(6,8)$ | $-2$ |
| $4$ | $2$ | $(1,2)(4,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,7,4,2)(3,6,5,8)$ | $0$ |
| $4$ | $4$ | $(1,3,4,5)(2,6,7,8)$ | $0$ |
| $2$ | $8$ | $(1,3,7,6,4,5,2,8)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,5,7,8,4,3,2,6)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
