Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 233 }$ to precision 11.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 42 + 232\cdot 233 + 59\cdot 233^{2} + 72\cdot 233^{3} + 35\cdot 233^{4} + 103\cdot 233^{5} + 98\cdot 233^{7} + 64\cdot 233^{8} + 90\cdot 233^{9} + 55\cdot 233^{10} +O\left(233^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 53 + 164\cdot 233 + 11\cdot 233^{2} + 186\cdot 233^{3} + 127\cdot 233^{4} + 206\cdot 233^{5} + 127\cdot 233^{6} + 114\cdot 233^{7} + 78\cdot 233^{8} + 48\cdot 233^{9} + 29\cdot 233^{10} +O\left(233^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 82 + 218\cdot 233 + 93\cdot 233^{2} + 75\cdot 233^{3} + 42\cdot 233^{4} + 55\cdot 233^{5} + 27\cdot 233^{6} + 42\cdot 233^{7} + 198\cdot 233^{8} + 232\cdot 233^{9} + 119\cdot 233^{10} +O\left(233^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 103 + 113\cdot 233 + 32\cdot 233^{2} + 134\cdot 233^{3} + 53\cdot 233^{4} + 65\cdot 233^{5} + 16\cdot 233^{6} + 182\cdot 233^{7} + 109\cdot 233^{8} + 130\cdot 233^{9} + 166\cdot 233^{10} +O\left(233^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 130 + 119\cdot 233 + 200\cdot 233^{2} + 98\cdot 233^{3} + 179\cdot 233^{4} + 167\cdot 233^{5} + 216\cdot 233^{6} + 50\cdot 233^{7} + 123\cdot 233^{8} + 102\cdot 233^{9} + 66\cdot 233^{10} +O\left(233^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 151 + 14\cdot 233 + 139\cdot 233^{2} + 157\cdot 233^{3} + 190\cdot 233^{4} + 177\cdot 233^{5} + 205\cdot 233^{6} + 190\cdot 233^{7} + 34\cdot 233^{8} + 113\cdot 233^{10} +O\left(233^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 180 + 68\cdot 233 + 221\cdot 233^{2} + 46\cdot 233^{3} + 105\cdot 233^{4} + 26\cdot 233^{5} + 105\cdot 233^{6} + 118\cdot 233^{7} + 154\cdot 233^{8} + 184\cdot 233^{9} + 203\cdot 233^{10} +O\left(233^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 191 + 173\cdot 233^{2} + 160\cdot 233^{3} + 197\cdot 233^{4} + 129\cdot 233^{5} + 232\cdot 233^{6} + 134\cdot 233^{7} + 168\cdot 233^{8} + 142\cdot 233^{9} + 177\cdot 233^{10} +O\left(233^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,7)(5,8)$ |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,3,8,6)(2,4,7,5)$ |
| $(1,5,8,4)(2,3,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,4)(2,7)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
| $4$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
| $2$ | $8$ | $(1,3,5,7,8,6,4,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,6,5,2,8,3,4,7)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
