Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 179 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 18 + 58\cdot 179 + 2\cdot 179^{2} + 27\cdot 179^{3} + 106\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 57 + 61\cdot 179 + 125\cdot 179^{2} + 172\cdot 179^{3} + 28\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 104 + 158\cdot 179 + 30\cdot 179^{2} + 89\cdot 179^{3} + 124\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 110 + 172\cdot 179 + 91\cdot 179^{2} + 12\cdot 179^{3} + 42\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 126 + 108\cdot 179 + 135\cdot 179^{2} + 22\cdot 179^{3} + 33\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 140 + 22\cdot 179 + 156\cdot 179^{2} + 165\cdot 179^{3} + 102\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 168 + 105\cdot 179 + 143\cdot 179^{2} + 67\cdot 179^{3} + 62\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 173 + 27\cdot 179 + 30\cdot 179^{2} + 158\cdot 179^{3} + 36\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,3,4,7,6,8,2)$ |
| $(1,6,7,5)(2,3,4,8)$ |
| $(1,8,7,3)(2,6,4,5)$ |
| $(1,7)(2,4)(3,8)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,4)(3,8)(5,6)$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,5)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,3,7,8)(2,5,4,6)$ | $0$ |
| $4$ | $4$ | $(1,5,7,6)(2,8,4,3)$ | $0$ |
| $2$ | $8$ | $(1,5,3,4,7,6,8,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,6,3,2,7,5,8,4)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
