Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 241 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 29 + 8\cdot 241 + 103\cdot 241^{2} + 185\cdot 241^{3} + 82\cdot 241^{4} + 63\cdot 241^{5} +O\left(241^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 + 6\cdot 241 + 78\cdot 241^{2} + 77\cdot 241^{3} + 84\cdot 241^{4} + 173\cdot 241^{5} +O\left(241^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 58 + 220\cdot 241 + 117\cdot 241^{2} + 199\cdot 241^{3} + 142\cdot 241^{4} + 31\cdot 241^{5} +O\left(241^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 112 + 114\cdot 241 + 112\cdot 241^{2} + 197\cdot 241^{3} + 124\cdot 241^{4} + 126\cdot 241^{5} +O\left(241^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 138 + 47\cdot 241 + 20\cdot 241^{2} + 2\cdot 241^{3} + 140\cdot 241^{4} + 210\cdot 241^{5} +O\left(241^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 177 + 165\cdot 241 + 96\cdot 241^{2} + 67\cdot 241^{3} + 108\cdot 241^{4} + 119\cdot 241^{5} +O\left(241^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 202 + 24\cdot 241 + 76\cdot 241^{2} + 109\cdot 241^{3} + 87\cdot 241^{4} + 98\cdot 241^{5} +O\left(241^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 221 + 135\cdot 241 + 118\cdot 241^{2} + 125\cdot 241^{3} + 193\cdot 241^{4} + 140\cdot 241^{5} +O\left(241^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,3,5,6,4,2,7)$ |
| $(2,3)(5,7)$ |
| $(4,8)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,6)(2,3)(4,8)(5,7)$ | $-4$ |
| $2$ | $2$ | $(4,8)(5,7)$ | $0$ |
| $4$ | $2$ | $(2,3)(5,7)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,6)(4,7)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,3,6,2)(4,7,8,5)$ | $0$ |
| $2$ | $4$ | $(1,3,6,2)(4,5,8,7)$ | $0$ |
| $4$ | $8$ | $(1,8,3,5,6,4,2,7)$ | $0$ |
| $4$ | $8$ | $(1,5,2,8,6,7,3,4)$ | $0$ |
| $4$ | $8$ | $(1,8,3,7,6,4,2,5)$ | $0$ |
| $4$ | $8$ | $(1,7,2,8,6,5,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
