Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 179 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 78\cdot 179 + 26\cdot 179^{2} + 85\cdot 179^{3} + 71\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 133\cdot 179 + 64\cdot 179^{2} + 123\cdot 179^{3} + 110\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 37 + 80\cdot 179 + 145\cdot 179^{2} + 65\cdot 179^{3} + 163\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 58 + 86\cdot 179 + 63\cdot 179^{2} + 157\cdot 179^{3} + 31\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 133 + 124\cdot 179 + 138\cdot 179^{2} + 53\cdot 179^{3} + 105\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 148 + 84\cdot 179 + 162\cdot 179^{2} + 101\cdot 179^{3} + 171\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 158 + 13\cdot 179 + 91\cdot 179^{2} + 23\cdot 179^{3} + 110\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 167 + 114\cdot 179 + 23\cdot 179^{2} + 105\cdot 179^{3} + 130\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,3,6)(2,4,5,7)$ |
| $(2,4)(5,7)(6,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
| $(1,2,8,4,3,5,6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $-2$ |
| $4$ | $2$ | $(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,8,3,6)(2,4,5,7)$ | $0$ |
| $4$ | $4$ | $(1,4,3,7)(2,6,5,8)$ | $0$ |
| $2$ | $8$ | $(1,2,8,4,3,5,6,7)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,5,8,7,3,2,6,4)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
