Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 179 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 41 + 26\cdot 179 + 76\cdot 179^{2} + 68\cdot 179^{3} + 51\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 53 + 53\cdot 179 + 137\cdot 179^{2} + 136\cdot 179^{3} + 130\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 79 + 73\cdot 179 + 82\cdot 179^{2} + 91\cdot 179^{3} + 175\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 88 + 78\cdot 179 + 35\cdot 179^{2} + 19\cdot 179^{3} + 103\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 91 + 100\cdot 179 + 143\cdot 179^{2} + 159\cdot 179^{3} + 75\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 100 + 105\cdot 179 + 96\cdot 179^{2} + 87\cdot 179^{3} + 3\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 126 + 125\cdot 179 + 41\cdot 179^{2} + 42\cdot 179^{3} + 48\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 138 + 152\cdot 179 + 102\cdot 179^{2} + 110\cdot 179^{3} + 127\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,2,6,4)(3,5,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,4)(3,8)(5,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,7)(4,5)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,8)(3,4)(5,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,6,4)(3,5,8,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
