Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 293 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 14 + 183\cdot 293 + 196\cdot 293^{2} + 211\cdot 293^{3} + 193\cdot 293^{4} + 172\cdot 293^{5} + 152\cdot 293^{6} +O\left(293^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 119 + 142\cdot 293 + 236\cdot 293^{2} + 139\cdot 293^{3} + 96\cdot 293^{4} + 67\cdot 293^{5} + 17\cdot 293^{6} +O\left(293^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 122 + 293 + 222\cdot 293^{2} + 269\cdot 293^{3} + 115\cdot 293^{4} + 173\cdot 293^{5} + 7\cdot 293^{6} +O\left(293^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 135 + 41\cdot 293 + 35\cdot 293^{2} + 103\cdot 293^{3} + 63\cdot 293^{4} + 63\cdot 293^{5} + 183\cdot 293^{6} +O\left(293^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 158 + 251\cdot 293 + 257\cdot 293^{2} + 189\cdot 293^{3} + 229\cdot 293^{4} + 229\cdot 293^{5} + 109\cdot 293^{6} +O\left(293^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 171 + 291\cdot 293 + 70\cdot 293^{2} + 23\cdot 293^{3} + 177\cdot 293^{4} + 119\cdot 293^{5} + 285\cdot 293^{6} +O\left(293^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 174 + 150\cdot 293 + 56\cdot 293^{2} + 153\cdot 293^{3} + 196\cdot 293^{4} + 225\cdot 293^{5} + 275\cdot 293^{6} +O\left(293^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 279 + 109\cdot 293 + 96\cdot 293^{2} + 81\cdot 293^{3} + 99\cdot 293^{4} + 120\cdot 293^{5} + 140\cdot 293^{6} +O\left(293^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,6,7,3)$ |
| $(1,7,8,2)$ |
| $(1,7,8,2)(3,5,6,4)$ |
| $(1,8)(2,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(1,8)(2,7)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,3)(2,4)(5,7)(6,8)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,7,8,2)(3,5,6,4)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,2,8,7)(3,4,6,5)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,7,8,2)$ |
$\zeta_{4} - 1$ |
$-\zeta_{4} - 1$ |
| $2$ |
$4$ |
$(1,2,8,7)$ |
$-\zeta_{4} - 1$ |
$\zeta_{4} - 1$ |
| $2$ |
$4$ |
$(1,8)(2,7)(3,5,6,4)$ |
$\zeta_{4} + 1$ |
$-\zeta_{4} + 1$ |
| $2$ |
$4$ |
$(1,8)(2,7)(3,4,6,5)$ |
$-\zeta_{4} + 1$ |
$\zeta_{4} + 1$ |
| $2$ |
$4$ |
$(1,2,8,7)(3,5,6,4)$ |
$0$ |
$0$ |
| $4$ |
$4$ |
$(1,4,8,5)(2,6,7,3)$ |
$0$ |
$0$ |
| $4$ |
$8$ |
$(1,4,2,6,8,5,7,3)$ |
$0$ |
$0$ |
| $4$ |
$8$ |
$(1,6,7,4,8,3,2,5)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
