Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 179 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 100\cdot 179 + 61\cdot 179^{2} + 88\cdot 179^{3} + 91\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 91\cdot 179 + 73\cdot 179^{2} + 61\cdot 179^{3} + 158\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 + 79\cdot 179 + 48\cdot 179^{2} + 72\cdot 179^{3} + 23\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 40 + 123\cdot 179 + 176\cdot 179^{2} + 20\cdot 179^{3} + 72\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 92 + 58\cdot 179 + 169\cdot 179^{2} + 39\cdot 179^{3} + 92\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 101 + 139\cdot 179 + 88\cdot 179^{2} + 44\cdot 179^{3} + 16\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 135 + 149\cdot 179 + 53\cdot 179^{2} + 62\cdot 179^{3} + 7\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 139 + 153\cdot 179 + 43\cdot 179^{2} + 147\cdot 179^{3} + 75\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6)(2,8)(3,4)(5,7)$ |
| $(4,5)(6,8)$ |
| $(1,4)(2,5)(3,8)(6,7)$ |
| $(1,2)(3,7)(4,5)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,7)(4,5)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,6)(2,8)(3,4)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,5)(3,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(4,5)(6,8)$ | $0$ |
| $1$ | $4$ | $(1,7,2,3)(4,6,5,8)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,3,2,7)(4,8,5,6)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,8,2,6)(3,5,7,4)$ | $0$ |
| $2$ | $4$ | $(1,7,2,3)(4,8,5,6)$ | $0$ |
| $2$ | $4$ | $(1,5,2,4)(3,6,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
