Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 179 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 51\cdot 179 + 70\cdot 179^{2} + 85\cdot 179^{3} + 176\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 + 131\cdot 179 + 77\cdot 179^{2} + 44\cdot 179^{3} + 68\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 45 + 36\cdot 179 + 133\cdot 179^{2} + 145\cdot 179^{3} + 139\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 70 + 18\cdot 179 + 111\cdot 179^{2} + 57\cdot 179^{3} + 67\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 111 + 8\cdot 179^{2} + 126\cdot 179^{3} + 110\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 146 + 47\cdot 179 + 83\cdot 179^{2} + 112\cdot 179^{3} + 60\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 148 + 32\cdot 179 + 59\cdot 179^{2} + 142\cdot 179^{3} + 81\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 163 + 39\cdot 179 + 173\cdot 179^{2} + 179^{3} + 11\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,7,2,8,3,6,4)$ |
| $(2,4)(3,5)$ |
| $(1,7,8,6)(2,5,4,3)$ |
| $(1,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,4)(3,5)(6,7)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(1,8)(6,7)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,7,8,6)(2,3,4,5)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,6,8,7)(2,5,4,3)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,7,8,6)(2,5,4,3)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,5,7,2,8,3,6,4)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,2,6,5,8,4,7,3)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,3,6,2,8,5,7,4)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,2,7,3,8,4,6,5)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
