Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 179 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 + 169\cdot 179 + 76\cdot 179^{2} + 130\cdot 179^{3} + 17\cdot 179^{4} + 65\cdot 179^{5} + 108\cdot 179^{6} +O\left(179^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 + 65\cdot 179 + 136\cdot 179^{2} + 29\cdot 179^{3} + 99\cdot 179^{4} + 92\cdot 179^{5} + 35\cdot 179^{6} +O\left(179^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 41 + 104\cdot 179 + 57\cdot 179^{2} + 69\cdot 179^{3} + 88\cdot 179^{4} + 89\cdot 179^{5} + 65\cdot 179^{6} +O\left(179^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 86 + 78\cdot 179 + 19\cdot 179^{2} + 37\cdot 179^{3} + 159\cdot 179^{4} + 114\cdot 179^{5} + 45\cdot 179^{6} +O\left(179^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 93 + 100\cdot 179 + 159\cdot 179^{2} + 141\cdot 179^{3} + 19\cdot 179^{4} + 64\cdot 179^{5} + 133\cdot 179^{6} +O\left(179^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 138 + 74\cdot 179 + 121\cdot 179^{2} + 109\cdot 179^{3} + 90\cdot 179^{4} + 89\cdot 179^{5} + 113\cdot 179^{6} +O\left(179^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 153 + 113\cdot 179 + 42\cdot 179^{2} + 149\cdot 179^{3} + 79\cdot 179^{4} + 86\cdot 179^{5} + 143\cdot 179^{6} +O\left(179^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 163 + 9\cdot 179 + 102\cdot 179^{2} + 48\cdot 179^{3} + 161\cdot 179^{4} + 113\cdot 179^{5} + 70\cdot 179^{6} +O\left(179^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(3,6)(4,5)$ |
| $(1,3,2,5,8,6,7,4)$ |
| $(1,2,8,7)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(3,6)(4,5)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,2,8,7)(3,5,6,4)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,7,8,2)(3,4,6,5)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,2,8,7)(3,4,6,5)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,3,2,5,8,6,7,4)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,5,7,3,8,4,2,6)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,5,2,6,8,4,7,3)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,6,7,5,8,3,2,4)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
