Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 179 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 42 + 54\cdot 179 + 111\cdot 179^{2} + 6\cdot 179^{3} + 165\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 46 + 76\cdot 179 + 151\cdot 179^{2} + 6\cdot 179^{3} + 114\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 49 + 122\cdot 179 + 153\cdot 179^{2} + 19\cdot 179^{3} + 124\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 50 + 156\cdot 179 + 2\cdot 179^{2} + 169\cdot 179^{3} + 73\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 101 + 155\cdot 179 + 92\cdot 179^{2} + 101\cdot 179^{3} + 139\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 136 + 98\cdot 179 + 164\cdot 179^{2} + 124\cdot 179^{3} + 100\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 142 + 113\cdot 179 + 82\cdot 179^{2} + 47\cdot 179^{3} + 177\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 153 + 117\cdot 179 + 135\cdot 179^{2} + 60\cdot 179^{3} +O\left(179^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,4,3)(2,5,7,8)$ |
| $(1,2,4,7)(3,5,6,8)$ |
| $(1,4)(2,7)(3,6)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,7)(3,6)(5,8)$ |
$-2$ |
| $2$ |
$4$ |
$(1,6,4,3)(2,5,7,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,4,7)(3,5,6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,8,4,5)(2,6,7,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
