Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 239 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 21 + 23\cdot 239 + 56\cdot 239^{2} + 86\cdot 239^{3} + 188\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 76 + 168\cdot 239 + 183\cdot 239^{2} + 61\cdot 239^{3} + 140\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 80 + 60\cdot 239 + 39\cdot 239^{2} + 26\cdot 239^{3} + 30\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 85 + 230\cdot 239 + 216\cdot 239^{2} + 57\cdot 239^{3} + 206\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 154 + 8\cdot 239 + 22\cdot 239^{2} + 181\cdot 239^{3} + 32\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 159 + 178\cdot 239 + 199\cdot 239^{2} + 212\cdot 239^{3} + 208\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 163 + 70\cdot 239 + 55\cdot 239^{2} + 177\cdot 239^{3} + 98\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 218 + 215\cdot 239 + 182\cdot 239^{2} + 152\cdot 239^{3} + 50\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,5)(4,7)(6,8)$ |
| $(1,2,8,7)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
