Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 241 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 68 + 74\cdot 241 + 20\cdot 241^{2} + 46\cdot 241^{3} + 121\cdot 241^{4} + 161\cdot 241^{5} + 78\cdot 241^{6} +O\left(241^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 73 + 195\cdot 241 + 13\cdot 241^{2} + 69\cdot 241^{3} + 31\cdot 241^{4} + 70\cdot 241^{5} + 75\cdot 241^{6} +O\left(241^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 94 + 155\cdot 241 + 111\cdot 241^{2} + 180\cdot 241^{3} + 58\cdot 241^{4} + 168\cdot 241^{5} + 12\cdot 241^{6} +O\left(241^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 110 + 104\cdot 241 + 30\cdot 241^{2} + 234\cdot 241^{3} + 120\cdot 241^{4} + 222\cdot 241^{5} + 143\cdot 241^{6} +O\left(241^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 191 + 231\cdot 241 + 77\cdot 241^{2} + 217\cdot 241^{3} + 106\cdot 241^{4} + 232\cdot 241^{5} + 110\cdot 241^{6} +O\left(241^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 212 + 147\cdot 241 + 78\cdot 241^{2} + 21\cdot 241^{3} + 181\cdot 241^{4} + 170\cdot 241^{5} + 5\cdot 241^{6} +O\left(241^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 228 + 187\cdot 241 + 213\cdot 241^{2} + 62\cdot 241^{3} + 135\cdot 241^{4} + 151\cdot 241^{5} + 111\cdot 241^{6} +O\left(241^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 233 + 107\cdot 241 + 176\cdot 241^{2} + 132\cdot 241^{3} + 208\cdot 241^{4} + 27\cdot 241^{5} + 184\cdot 241^{6} +O\left(241^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,8)(3,6)(5,7)$ |
| $(2,8)(3,6)$ |
| $(1,2,5,6,4,8,7,3)$ |
| $(1,5,4,7)(2,3,8,6)$ |
| $(2,3,8,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,8)(3,6)(5,7)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(2,8)(3,6)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,2)(3,7)(4,8)(5,6)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,5,4,7)(2,6,8,3)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,7,4,5)(2,3,8,6)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,5,4,7)(2,3,8,6)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(2,3,8,6)$ |
$-\zeta_{4} - 1$ |
$\zeta_{4} - 1$ |
| $2$ |
$4$ |
$(2,6,8,3)$ |
$\zeta_{4} - 1$ |
$-\zeta_{4} - 1$ |
| $2$ |
$4$ |
$(1,5,4,7)(2,8)(3,6)$ |
$\zeta_{4} + 1$ |
$-\zeta_{4} + 1$ |
| $2$ |
$4$ |
$(1,7,4,5)(2,8)(3,6)$ |
$-\zeta_{4} + 1$ |
$\zeta_{4} + 1$ |
| $4$ |
$4$ |
$(1,6,4,3)(2,5,8,7)$ |
$0$ |
$0$ |
| $4$ |
$8$ |
$(1,2,5,6,4,8,7,3)$ |
$0$ |
$0$ |
| $4$ |
$8$ |
$(1,6,7,2,4,3,5,8)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
