Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 239 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 19 + 103\cdot 239 + 135\cdot 239^{2} + 94\cdot 239^{3} + 168\cdot 239^{4} + 214\cdot 239^{5} +O\left(239^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 + 22\cdot 239 + 177\cdot 239^{2} + 211\cdot 239^{3} + 20\cdot 239^{4} + 206\cdot 239^{5} +O\left(239^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 87 + 41\cdot 239 + 15\cdot 239^{2} + 178\cdot 239^{3} + 223\cdot 239^{4} + 176\cdot 239^{5} +O\left(239^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 93 + 199\cdot 239 + 56\cdot 239^{2} + 56\cdot 239^{3} + 76\cdot 239^{4} + 168\cdot 239^{5} +O\left(239^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 149 + 57\cdot 239 + 11\cdot 239^{2} + 112\cdot 239^{3} + 46\cdot 239^{4} + 118\cdot 239^{5} +O\left(239^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 176 + 40\cdot 239 + 239^{2} + 208\cdot 239^{3} + 26\cdot 239^{4} + 51\cdot 239^{5} +O\left(239^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 191 + 134\cdot 239 + 45\cdot 239^{2} + 119\cdot 239^{3} + 206\cdot 239^{4} + 43\cdot 239^{5} +O\left(239^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 218 + 117\cdot 239 + 35\cdot 239^{2} + 215\cdot 239^{3} + 186\cdot 239^{4} + 215\cdot 239^{5} +O\left(239^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,4,2)(5,6,8,7)$ |
| $(1,6,2,8,4,7,3,5)$ |
| $(1,4)(2,3)(5,8)(6,7)$ |
| $(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(5,8)(6,7)$ | $0$ |
| $1$ | $4$ | $(1,2,4,3)(5,6,8,7)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,3,4,2)(5,7,8,6)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,4,2)(5,6,8,7)$ | $0$ |
| $2$ | $8$ | $(1,6,2,8,4,7,3,5)$ | $0$ |
| $2$ | $8$ | $(1,8,3,6,4,5,2,7)$ | $0$ |
| $2$ | $8$ | $(1,8,2,7,4,5,3,6)$ | $0$ |
| $2$ | $8$ | $(1,7,3,8,4,6,2,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
