Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 271 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 117\cdot 271 + 190\cdot 271^{2} + 24\cdot 271^{3} + 53\cdot 271^{4} +O\left(271^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 108\cdot 271 + 236\cdot 271^{2} + 18\cdot 271^{3} + 177\cdot 271^{4} +O\left(271^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 63 + 208\cdot 271 + 142\cdot 271^{2} + 4\cdot 271^{3} + 75\cdot 271^{4} +O\left(271^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 121 + 94\cdot 271 + 129\cdot 271^{2} + 234\cdot 271^{3} + 137\cdot 271^{4} +O\left(271^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 154 + 121\cdot 271 + 68\cdot 271^{2} + 136\cdot 271^{3} + 138\cdot 271^{4} +O\left(271^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 237 + 61\cdot 271 + 147\cdot 271^{2} + 187\cdot 271^{3} + 34\cdot 271^{4} +O\left(271^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 241 + 225\cdot 271 + 183\cdot 271^{2} + 260\cdot 271^{3} + 150\cdot 271^{4} +O\left(271^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 246 + 146\cdot 271 + 256\cdot 271^{2} + 216\cdot 271^{3} + 45\cdot 271^{4} +O\left(271^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,8,6,7,5,3,4)$ |
| $(2,5)(4,6)$ |
| $(1,8,7,3)(2,4,5,6)$ |
| $(1,7)(2,5)(3,8)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,5)(3,8)(4,6)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(2,5)(4,6)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,8,7,3)(2,6,5,4)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,3,7,8)(2,4,5,6)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,8,7,3)(2,4,5,6)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,2,8,6,7,5,3,4)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,6,3,2,7,4,8,5)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,6,8,5,7,4,3,2)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,5,3,6,7,2,8,4)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
