Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 233 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 27 + 35\cdot 233 + 132\cdot 233^{2} + 37\cdot 233^{3} + 144\cdot 233^{4} +O\left(233^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 36 + 154\cdot 233^{2} + 132\cdot 233^{3} + 99\cdot 233^{4} +O\left(233^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 78 + 158\cdot 233 + 34\cdot 233^{2} + 192\cdot 233^{3} + 116\cdot 233^{4} +O\left(233^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 92 + 39\cdot 233 + 145\cdot 233^{2} + 103\cdot 233^{3} + 105\cdot 233^{4} +O\left(233^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 141 + 193\cdot 233 + 87\cdot 233^{2} + 129\cdot 233^{3} + 127\cdot 233^{4} +O\left(233^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 155 + 74\cdot 233 + 198\cdot 233^{2} + 40\cdot 233^{3} + 116\cdot 233^{4} +O\left(233^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 197 + 232\cdot 233 + 78\cdot 233^{2} + 100\cdot 233^{3} + 133\cdot 233^{4} +O\left(233^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 206 + 197\cdot 233 + 100\cdot 233^{2} + 195\cdot 233^{3} + 88\cdot 233^{4} +O\left(233^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,2,4)(5,8,6,7)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,6)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,3,2,4)(5,8,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
