Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 241 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 28\cdot 241 + 137\cdot 241^{2} + 157\cdot 241^{3} + 222\cdot 241^{4} + 164\cdot 241^{5} + 231\cdot 241^{6} +O\left(241^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 + 215\cdot 241 + 191\cdot 241^{2} + 140\cdot 241^{3} + 193\cdot 241^{4} + 186\cdot 241^{5} + 71\cdot 241^{6} +O\left(241^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 40 + 134\cdot 241 + 12\cdot 241^{2} + 143\cdot 241^{3} + 38\cdot 241^{4} + 113\cdot 241^{5} + 15\cdot 241^{6} +O\left(241^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 56 + 25\cdot 241 + 27\cdot 241^{2} + 133\cdot 241^{3} + 97\cdot 241^{4} + 70\cdot 241^{5} + 223\cdot 241^{6} +O\left(241^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 185 + 215\cdot 241 + 213\cdot 241^{2} + 107\cdot 241^{3} + 143\cdot 241^{4} + 170\cdot 241^{5} + 17\cdot 241^{6} +O\left(241^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 201 + 106\cdot 241 + 228\cdot 241^{2} + 97\cdot 241^{3} + 202\cdot 241^{4} + 127\cdot 241^{5} + 225\cdot 241^{6} +O\left(241^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 223 + 25\cdot 241 + 49\cdot 241^{2} + 100\cdot 241^{3} + 47\cdot 241^{4} + 54\cdot 241^{5} + 169\cdot 241^{6} +O\left(241^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 238 + 212\cdot 241 + 103\cdot 241^{2} + 83\cdot 241^{3} + 18\cdot 241^{4} + 76\cdot 241^{5} + 9\cdot 241^{6} +O\left(241^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,7,8,2)(3,5,6,4)$ |
| $(3,6)(4,5)$ |
| $(1,4,2,3,8,5,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $2$ | $8$ | $(1,4,2,3,8,5,7,6)$ | $0$ |
| $2$ | $8$ | $(1,3,7,4,8,6,2,5)$ | $0$ |
| $2$ | $8$ | $(1,3,2,5,8,6,7,4)$ | $0$ |
| $2$ | $8$ | $(1,5,7,3,8,4,2,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
