Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 179 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 117\cdot 179 + 22\cdot 179^{2} + 40\cdot 179^{3} + 18\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 173\cdot 179 + 61\cdot 179^{2} + 18\cdot 179^{3} + 145\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 37 + 100\cdot 179 + 100\cdot 179^{2} + 28\cdot 179^{3} + 144\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 49 + 85\cdot 179 + 147\cdot 179^{2} + 94\cdot 179^{3} + 17\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 58 + 70\cdot 179 + 31\cdot 179^{2} + 47\cdot 179^{3} + 55\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 87 + 134\cdot 179 + 169\cdot 179^{2} + 72\cdot 179^{3} + 68\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 135 + 58\cdot 179 + 43\cdot 179^{2} + 131\cdot 179^{3} + 52\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 155 + 155\cdot 179 + 138\cdot 179^{2} + 103\cdot 179^{3} + 35\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,4)(3,6)(5,7)$ |
| $(1,7,4,2,5,8,3,6)$ |
| $(1,3,5,4)(2,7,6,8)$ |
| $(1,5)(2,6)(3,4)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,4)(7,8)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,8)(2,4)(3,6)(5,7)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,5)(2,7)(6,8)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,3,5,4)(2,7,6,8)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,6,3,8,5,2,4,7)$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ |
$8$ |
$(1,8,4,6,5,7,3,2)$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
