Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 179 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 9\cdot 179 + 153\cdot 179^{2} + 106\cdot 179^{3} + 52\cdot 179^{4} + 127\cdot 179^{5} +O\left(179^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 85\cdot 179 + 70\cdot 179^{2} + 33\cdot 179^{3} + 107\cdot 179^{4} + 129\cdot 179^{5} +O\left(179^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 + 139\cdot 179 + 53\cdot 179^{2} + 19\cdot 179^{3} + 171\cdot 179^{4} + 140\cdot 179^{5} +O\left(179^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 + 72\cdot 179 + 102\cdot 179^{2} + 104\cdot 179^{3} + 101\cdot 179^{4} + 83\cdot 179^{5} +O\left(179^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 87 + 61\cdot 179 + 131\cdot 179^{2} + 21\cdot 179^{3} + 157\cdot 179^{4} + 3\cdot 179^{5} +O\left(179^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 108 + 137\cdot 179 + 48\cdot 179^{2} + 127\cdot 179^{3} + 32\cdot 179^{4} + 6\cdot 179^{5} +O\left(179^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 113 + 101\cdot 179 + 28\cdot 179^{2} + 38\cdot 179^{3} + 14\cdot 179^{4} + 128\cdot 179^{5} +O\left(179^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 136 + 109\cdot 179 + 127\cdot 179^{2} + 85\cdot 179^{3} + 79\cdot 179^{4} + 96\cdot 179^{5} +O\left(179^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,5)(3,8)(4,7)$ |
| $(1,2,6,5)(3,8,4,7)$ |
| $(1,4,5,8,6,3,2,7)$ |
| $(1,6)(2,5)(3,4)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ | $-2$ |
| $4$ | $2$ | $(2,5)(3,8)(4,7)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,8)(3,6)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,5,6,2)(3,7,4,8)$ | $0$ |
| $2$ | $8$ | $(1,4,5,8,6,3,2,7)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,8,2,4,6,7,5,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
