Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 233 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 40 + 40\cdot 233 + 161\cdot 233^{2} + 203\cdot 233^{3} + 128\cdot 233^{4} +O\left(233^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 50 + 75\cdot 233 + 98\cdot 233^{2} + 101\cdot 233^{3} + 63\cdot 233^{4} +O\left(233^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 51 + 193\cdot 233 + 39\cdot 233^{2} + 43\cdot 233^{3} + 58\cdot 233^{4} +O\left(233^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 94 + 157\cdot 233 + 166\cdot 233^{2} + 117\cdot 233^{3} + 215\cdot 233^{4} +O\left(233^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 135 + 57\cdot 233 + 188\cdot 233^{2} + 132\cdot 233^{3} + 174\cdot 233^{4} +O\left(233^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 179 + 139\cdot 233 + 23\cdot 233^{2} + 149\cdot 233^{3} + 93\cdot 233^{4} +O\left(233^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 188 + 57\cdot 233 + 71\cdot 233^{2} + 172\cdot 233^{3} + 17\cdot 233^{4} +O\left(233^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 199 + 210\cdot 233 + 182\cdot 233^{2} + 11\cdot 233^{3} + 180\cdot 233^{4} +O\left(233^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,8)(4,6)(5,7)$ |
| $(1,3)(2,4)(5,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,7)(3,6)(4,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,8)(4,6)(5,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,4)(5,6)(7,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,5,8)(2,3,7,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
