Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 179 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 24 + 27\cdot 179 + 95\cdot 179^{2} + 112\cdot 179^{3} + 63\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 + 160\cdot 179 + 28\cdot 179^{2} + 34\cdot 179^{3} + 10\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 57 + 145\cdot 179 + 167\cdot 179^{2} + 138\cdot 179^{3} + 177\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 103 + 99\cdot 179 + 97\cdot 179^{2} + 28\cdot 179^{3} + 118\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 164 + 162\cdot 179 + 101\cdot 179^{2} + 155\cdot 179^{3} + 59\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 170 + 131\cdot 179 + 63\cdot 179^{2} + 156\cdot 179^{3} + 51\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 174 + 85\cdot 179 + 176\cdot 179^{2} + 77\cdot 179^{3} + 177\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 175 + 81\cdot 179 + 163\cdot 179^{2} + 11\cdot 179^{3} + 57\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,8)(6,7)$ |
| $(1,3,7,4)(2,8,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,6)(3,4)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,3)(4,6)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,3,7,4)(2,8,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
