Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 241 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 62 + 213\cdot 241 + 91\cdot 241^{2} + 9\cdot 241^{3} + 240\cdot 241^{4} + 193\cdot 241^{5} +O\left(241^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 76 + 112\cdot 241 + 193\cdot 241^{2} + 83\cdot 241^{3} + 136\cdot 241^{4} + 54\cdot 241^{5} +O\left(241^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 110 + 186\cdot 241 + 98\cdot 241^{2} + 63\cdot 241^{3} + 77\cdot 241^{4} + 218\cdot 241^{5} +O\left(241^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 117 + 22\cdot 241 + 94\cdot 241^{2} + 108\cdot 241^{3} + 64\cdot 241^{4} + 37\cdot 241^{5} +O\left(241^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 145 + 92\cdot 241 + 27\cdot 241^{2} + 241^{3} + 118\cdot 241^{4} +O\left(241^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 148 + 143\cdot 241 + 18\cdot 241^{2} + 6\cdot 241^{3} + 215\cdot 241^{4} + 120\cdot 241^{5} +O\left(241^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 149 + 39\cdot 241 + 171\cdot 241^{2} + 87\cdot 241^{3} + 53\cdot 241^{4} + 88\cdot 241^{5} +O\left(241^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 159 + 153\cdot 241 + 27\cdot 241^{2} + 122\cdot 241^{3} + 59\cdot 241^{4} + 9\cdot 241^{5} +O\left(241^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(5,8)(6,7)$ |
| $(2,3)(5,8)$ |
| $(1,4)(6,7)$ |
| $(1,2,5,7,4,3,8,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-4$ |
| $2$ | $2$ | $(2,3)(6,7)$ | $0$ |
| $4$ | $2$ | $(2,3)(5,8)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,5,4,8)(2,7,3,6)$ | $0$ |
| $2$ | $4$ | $(1,8,4,5)(2,7,3,6)$ | $0$ |
| $4$ | $8$ | $(1,2,5,7,4,3,8,6)$ | $0$ |
| $4$ | $8$ | $(1,7,8,2,4,6,5,3)$ | $0$ |
| $4$ | $8$ | $(1,2,8,7,4,3,5,6)$ | $0$ |
| $4$ | $8$ | $(1,7,5,2,4,6,8,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
