Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 139 }$ to precision 12.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 114\cdot 139 + 93\cdot 139^{2} + 18\cdot 139^{3} + 33\cdot 139^{4} + 22\cdot 139^{5} + 73\cdot 139^{6} + 96\cdot 139^{7} + 21\cdot 139^{8} + 82\cdot 139^{9} + 25\cdot 139^{10} + 113\cdot 139^{11} +O\left(139^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 + 69\cdot 139 + 27\cdot 139^{2} + 72\cdot 139^{3} + 86\cdot 139^{4} + 21\cdot 139^{5} + 127\cdot 139^{6} + 66\cdot 139^{7} + 64\cdot 139^{8} + 19\cdot 139^{9} + 91\cdot 139^{10} + 3\cdot 139^{11} +O\left(139^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 29 + 139 + 69\cdot 139^{2} + 72\cdot 139^{3} + 83\cdot 139^{4} + 45\cdot 139^{5} + 18\cdot 139^{6} + 11\cdot 139^{7} + 34\cdot 139^{8} + 134\cdot 139^{9} + 132\cdot 139^{10} + 5\cdot 139^{11} +O\left(139^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 58 + 31\cdot 139 + 133\cdot 139^{2} + 13\cdot 139^{3} + 132\cdot 139^{4} + 16\cdot 139^{5} + 104\cdot 139^{6} + 13\cdot 139^{7} + 7\cdot 139^{8} + 53\cdot 139^{9} + 53\cdot 139^{10} + 137\cdot 139^{11} +O\left(139^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 81 + 107\cdot 139 + 5\cdot 139^{2} + 125\cdot 139^{3} + 6\cdot 139^{4} + 122\cdot 139^{5} + 34\cdot 139^{6} + 125\cdot 139^{7} + 131\cdot 139^{8} + 85\cdot 139^{9} + 85\cdot 139^{10} + 139^{11} +O\left(139^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 110 + 137\cdot 139 + 69\cdot 139^{2} + 66\cdot 139^{3} + 55\cdot 139^{4} + 93\cdot 139^{5} + 120\cdot 139^{6} + 127\cdot 139^{7} + 104\cdot 139^{8} + 4\cdot 139^{9} + 6\cdot 139^{10} + 133\cdot 139^{11} +O\left(139^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 129 + 69\cdot 139 + 111\cdot 139^{2} + 66\cdot 139^{3} + 52\cdot 139^{4} + 117\cdot 139^{5} + 11\cdot 139^{6} + 72\cdot 139^{7} + 74\cdot 139^{8} + 119\cdot 139^{9} + 47\cdot 139^{10} + 135\cdot 139^{11} +O\left(139^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 135 + 24\cdot 139 + 45\cdot 139^{2} + 120\cdot 139^{3} + 105\cdot 139^{4} + 116\cdot 139^{5} + 65\cdot 139^{6} + 42\cdot 139^{7} + 117\cdot 139^{8} + 56\cdot 139^{9} + 113\cdot 139^{10} + 25\cdot 139^{11} +O\left(139^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,6)(3,7)(4,5)$ |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,6,8,3)(2,4,7,5)$ |
| $(1,4,8,5)(2,3,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(2,6)(3,7)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
| $4$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
| $2$ | $8$ | $(1,2,5,6,8,7,4,3)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,7,5,3,8,2,4,6)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
